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9019.Gerd Baumann - Mathematica for Theoretical Physics- Electrodynamics Quantum Mechanics General Relativity and Fractals (2005 Springer).pdf

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Mathematica╝ for Theoretical Physics
╝
Mathematica
for Theoretical Physics
Electrodynamics,
Quantum Mechanics,
General Relativity,
and Fractals
Second Edition
Gerd Baumann
CD-ROM Included
Gerd Baumann
Department of Mathematics
German University in Cairo GUC
New Cairo City
Main Entrance of Al Tagamoa Al Khames
Egypt
Gerd.Baumann@GUC.edu.eg
This is a translated, expanded, and updated version of the original German version of
the work ?Mathematica╝ in der Theoretischen Physik,? published by Springer-Verlag
Heidelberg, 1993 ╘.
Library of Congress Cataloging-in-Publication Data
Baumann, Gerd.
[Mathematica in der theoretischen Physik. English]
Mathematica for theoretical physics / by Gerd Baumann.?2nd ed.
p. cm.
Includes bibliographical references and index.
Contents: 1. Classical mechanics and nonlinear dynamics ? 2. Electrodynamics, quantum
mechanics, general relativity, and fractals.
ISBN 0-387-21933-1
1. Mathematical physics?Data processing. 2. Mathematica (Computer file) I. Title.
QC20.7.E4B3813 2004
530?.285?53?dc22
ISBN-10: 0-387-21933-1
ISBN-13: 978-0387-21933-2
2004046861
e-ISBN 0-387-25113-8
Printed on acid-free paper.
╘ 2005 Springer Science+Business Media, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New
York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis.
Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if
they are not identified as such, is not to be taken as an expression of opinion as to whether or not
they are subject to proprietary rights.
Mathematica, MathLink, and Math Source are registered trademarks of Wolfram Research, Inc.
Printed in the United States of America.
9 8 7 6 5 4 3 2 1
springeronline.com
(HAM)
To Carin,
for her love, support, and encuragement.
Preface
As physicists, mathematicians or engineers, we are all involved with
mathematical calculations in our everyday work. Most of the laborious,
complicated, and time-consuming calculations have to be done over and
over again if we want to check the validity of our assumptions and
derive new phenomena from changing models. Even in the age of
computers, we often use paper and pencil to do our calculations.
However, computer programs like Mathematica have revolutionized our
working methods. Mathematica not only supports popular numerical
calculations but also enables us to do exact analytical calculations by
computer. Once we know the analytical representations of physical
phenomena, we are able to use Mathematica to create graphical
representations of these relations. Days of calculations by hand have
shrunk to minutes by using Mathematica. Results can be verified within
a few seconds, a task that took hours if not days in the past.
The present text uses Mathematica as a tool to discuss and to solve
examples from physics. The intention of this book is to demonstrate the
usefulness of Mathematica in everyday applications. We will not give a
complete description of its syntax but demonstrate by examples the use
of its language. In particular, we show how this modern tool is used to
solve classical problems.
viii
Preface
This second edition of Mathematica in Theoretical Physics seeks to
prevent the objectives and emphasis of the previous edition. It is
extended to include a full course in classical mechanics, new examples
in quantum mechanics, and measurement methods for fractals. In
addition, there is an extension of the fractal's chapter by a fractional
calculus. The additional material and examples enlarged the text so
much that we decided to divide the book in two volumes. The first
volume covers classical mechanics and nonlinear dynamics. The second
volume starts with electrodynamics, adds quantum mechanics and
general relativity, and ends with fractals. Because of the inclusion of
new materials, it was necessary to restructure the text. The main
differences are concerned with the chapter on nonlinear dynamics. This
chapter discusses mainly classical field theory and, thus, it was
appropriate to locate it in line with the classical mechanics chapter.
The text contains a large number of examples that are solvable using
Mathematica. The defined functions and packages are available on CD
accompanying each of the two volumes. The names of the files on the
CD carry the names of their respective chapters. Chapter 1 comments on
the basic properties of Mathematica using examples from different fields
of physics. Chapter 2 demonstrates the use of Mathematica in a
step-by-step procedure applied to mechanical problems. Chapter 2
contains a one-term lecture in mechanics. It starts with the basic
definitions, goes on with Newton's mechanics, discusses the Lagrange
and Hamilton representation of mechanics, and ends with the rigid body
motion. We show how Mathematica is used to simplify our work and to
support and derive solutions for specific problems. In Chapter 3, we
examine nonlinear phenomena of the Korteweg?de Vries equation. We
demonstrate that Mathematica is an appropriate tool to derive numerical
and analytical solutions even for nonlinear equations of motion. The
second volume starts with Chapter 4, discussing problems of
electrostatics and the motion of ions in an electromagnetic field. We
further introduce Mathematica functions that are closely related to the
theoretical considerations of the selected problems. In Chapter 5, we
discuss problems of quantum mechanics. We examine the dynamics of a
free particle by the example of the time-dependent SchrЖdinger equation
and study one-dimensional eigenvalue problems using the analytic and
Preface
ix
numeric capabilities of Mathematica. Problems of general relativity are
discussed in Chapter 6. Most standard books on Einstein's theory discuss
the phenomena of general relativity by using approximations. With
Mathematica, general relativity effects like the shift of the perihelion
can be tracked with precision. Finally, the last chapter, Chapter 7, uses
computer algebra to represent fractals and gives an introduction to the
spatial renormalization theory. In addition, we present the basics of
fractional calculus approaching fractals from the analytic side. This
approach is supported by a package, FractionalCalculus, which is not
included in this project. The package is available by request from the
author. Exercises with which Mathematica can be used for modified
applications. Chapters 2?7 include at the end some exercises allowing
the reader to carry out his own experiments with the book.
Acknowledgments Since the first printing of this text, many people
made valuable contributions and gave excellent input. Because the
number of responses are so numerous, I give my thanks to all who
contributed by remarks and enhancements to the text. Concerning the
historical pictures used in the text, I acknowledge the support of the
http://www-gapdcs.st-and.ac.uk/~history/ webserver of the University of
St Andrews, Scotland. My special thanks go to Norbert SЭdland, who
made the package FractionalCalculus available for this text. I'm also
indebted to Hans KЖlsch and Virginia Lipscy, Springer-Verlag New
York Physics editorial. Finally, the author deeply appreciates the
understanding and support of his wife, Carin, and daughter, Andrea,
during the preparation of the book.
Cairo, Spring 2005
Gerd Baumann
Contents
Volume I
1
2
Preface
Introduction
1.1
Basics
1.1.1
1.1.2
1.1.3
1.1.4
1.1.5
1.1.6
Structure of Mathematica
Interactive Use of Mathematica
Symbolic Calculations
Numerical Calculations
Graphics
Programming
Classical Mechanics
2.1
Introduction
2.2
Mathematical Tools
2.2.1 Introduction
2.2.2 Coordinates
2.2.3 Coordinate Transformations and Matrices
2.2.4 Scalars
2.2.5 Vectors
2.2.6 Tensors
2.2.7 Vector Products
2.2.8 Derivatives
2.2.9 Integrals
2.2.10 Exercises
vii
1
1
2
4
6
11
13
23
31
31
35
35
36
38
54
57
59
64
69
73
74
xii
Contents
2.3
2.4
2.5
2.6
2.7
Kinematics
2.3.1 Introduction
2.3.2 Velocity
2.3.3 Acceleration
2.3.4 Kinematic Examples
2.3.5 Exercises
Newtonian Mechanics
2.4.1 Introduction
2.4.2 Frame of Reference
2.4.3 Time
2.4.4 Mass
2.4.5 Newton's Laws
2.4.6 Forces in Nature
2.4.7 Conservation Laws
2.4.8 Application of Newton's Second Law
2.4.9 Exercises
2.4.10 Packages and Programs
Central Forces
2.5.1 Introduction
2.5.2 Kepler's Laws
2.5.3 Central Field Motion
2.5.4 Two-Particle Collisons and Scattering
2.5.5 Exercises
2.5.6 Packages and Programs
Calculus of Variations
2.6.1 Introduction
2.6.2 The Problem of Variations
2.6.3 Euler's Equation
2.6.4 Euler Operator
2.6.5 Algorithm Used in the Calculus of Variations
2.6.6 Euler Operator for q Dependent Variables
2.6.7 Euler Operator for q + p Dimensions
2.6.8 Variations with Constraints
2.6.9 Exercises
2.6.10 Packages and Programs
Lagrange Dynamics
2.7.1 Introduction
2.7.2 Hamilton's Principle Hisorical Remarks
76
76
77
81
82
94
96
96
98
100
101
103
106
111
118
188
188
201
201
202
208
240
272
273
274
274
276
281
283
284
293
296
300
303
303
305
305
306
Contents
xiii
2.8
2.9
2.10
3
2.7.3 Hamilton's Principle
2.7.4 Symmetries and Conservation Laws
2.7.5 Exercises
2.7.6 Packages and Programs
Hamiltonian Dynamics
2.8.1 Introduction
2.8.2 Legendre Transform
2.8.3 Hamilton's Equation of Motion
2.8.4 Hamilton's Equations and the Calculus of Variation
2.8.5 Liouville's Theorem
2.8.6 Poisson Brackets
2.8.7 Manifolds and Classes
2.8.8 Canonical Transformations
2.8.9 Generating Functions
2.8.10 Action Variables
2.8.11 Exercises
2.8.12 Packages and Programs
Chaotic Systems
2.9.1 Introduction
2.9.2 Discrete Mappings and Hamiltonians
2.9.3 Lyapunov Exponents
2.9.4 Exercises
Rigid Body
2.10.1 Introduction
2.10.2 The Inertia Tensor
2.10.3 The Angular Momentum
2.10.4 Principal Axes of Inertia
2.10.5 Steiner's Theorem
2.10.6 Euler's Equations of Motion
2.10.7 Force-Free Motion of a Symmetrical Top
2.10.8 Motion of a Symmetrical Top in a Force Field
2.10.9 Exercises
2.10.10 Packages and Programms
Nonlinear Dynamics
3.1
Introduction
3.2
The Korteweg?de Vries Equation
3.3
Solution of the Korteweg-de Vries Equation
313
341
351
351
354
354
355
362
366
373
377
384
396
398
403
419
419
422
422
431
435
448
449
449
450
453
454
460
462
467
471
481
481
485
485
488
492
xiv
Contents
3.3.1
3.3.2
3.4
3.5
3.6
3.7
The Inverse Scattering Transform
Soliton Solutions of the Korteweg?de Vries
Equation
Conservation Laws of the Korteweg?de Vries Equation
3.4.1 Definition of Conservation Laws
3.4.2 Derivation of Conservation Laws
Numerical Solution of the Korteweg?de Vries Equation
Exercises
Packages and Programs
3.7.1 Solution of the KdV Equation
3.7.2 Conservation Laws for the KdV Equation
3.7.3 Numerical Solution of the KdV Equation
References
Index
492
498
505
506
508
511
515
516
516
517
518
521
529
Volume II
4
5
Preface
Electrodynamics
4.1
Introduction
4.2
Potential and Electric Field of Discrete Charge
Distributions
4.3
Boundary Problem of Electrostatics
4.4
Two Ions in the Penning Trap
4.4.1 The Center of Mass Motion
4.4.2 Relative Motion of the Ions
4.5
Exercises
4.6
Packages and Programs
4.6.1 Point Charges
4.6.2 Boundary Problem
4.6.3 Penning Trap
vii
545
545
Quantum Mechanics
5.1
Introduction
5.2
The SchrЖdinger Equation
587
587
590
548
555
566
569
572
577
578
578
581
582
Contents
xv
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6
One-Dimensional Potential
The Harmonic Oscillator
Anharmonic Oscillator
Motion in the Central Force Field
Second Virial Coefficient and Its Quantum Corrections
5.7.1 The SVC and Its Relation to Thermodynamic
Properties
5.7.2 Calculation of the Classical SVC Bc HTL for the
H2 n - nL -Potential
5.7.3 Quantum Mechanical Corrections Bq1 HTL and
Bq2 HTL of the SVC
5.7.4 Shape Dependence of the Boyle Temperature
5.7.5 The High-Temperature Partition Function for
Diatomic Molecules
Exercises
Packages and Programs
5.9.1 QuantumWell
5.9.2 HarmonicOscillator
5.9.3 AnharmonicOscillator
5.9.4 CentralField
595
609
619
631
642
644
646
655
680
684
687
688
688
693
695
698
General Relativity
703
6.1
Introduction
703
6.2
The Orbits in General Relativity
707
6.2.1 Quasielliptic Orbits
713
6.2.2 Asymptotic Circles
719
6.3
Light Bending in the Gravitational Field
720
6.4
Einstein's Field Equations (Vacuum Case)
725
6.4.1 Examples for Metric Tensors
727
6.4.2 The Christoffel Symbols
731
6.4.3 The Riemann Tensor
731
6.4.4 Einstein's Field Equations
733
6.4.5 The Cartesian Space
734
6.4.6 Cartesian Space in Cylindrical Coordinates
736
6.4.7 Euclidean Space in Polar Coordinates
737
6.5
The Schwarzschild Solution
739
6.5.1 The Schwarzschild Metric in Eddington?Finkelstein
Form
739
xvi
Contents
6.6
6.7
6.8
7
6.5.2 Dingle's Metric
6.5.3 Schwarzschild Metric in Kruskal Coordinates
The Reissner?Nordstrom Solution for a Charged
Mass Point
Exercises
Packages and Programs
6.8.1 EulerLagrange Equations
6.8.2 PerihelionShift
6.8.3 LightBending
742
748
752
759
761
761
762
767
Fractals
7.1
Introduction
7.2
Measuring a Borderline
7.2.1 Box Counting
7.3
The Koch Curve
7.4
Multifractals
7.4.1 Multifractals with Common Scaling Factor
7.5
The Renormlization Group
7.6
Fractional Calculus
7.6.1 Historical Remarks on Fractional Calculus
7.6.2 The Riemann?Liouville Calculus
7.6.3 Mellin Transforms
7.6.4 Fractional Differential Equations
7.7
Exercises
7.8
Packages and Programs
7.8.1 Tree Generation
7.8.2 Koch Curves
7.8.3 Multifactals
7.8.4 Renormalization
7.8.5 Fractional Calculus
773
773
776
781
790
795
798
801
809
810
813
830
856
883
883
883
886
892
895
897
Appendix
A.1
Program Installation
A.2
Glossary of Files and Functions
A.3
Mathematica Functions
899
899
900
910
References
Index
923
931
4
Electrodynamics
4.1 Introduction
This chapter is concerned with electric fields and charges encountered in
different systems. Electricity is an ancient phenomenon already known by
the Greeks. The experimental and theoretical basis of the current
understanding of electrodynamical phenomena was established by two
men: Michael Farady, the self-trained experimenter, and James Clerk
Maxwell, the theoretician. The work of both were based on extensive
material and knowledge by Coulomb. Farady, originally, a bookbinder,
was most interested in electricity. His inquisitiveness in gaining
knowledge on electrical phenomena made it possible to obtain an
assistantship in Davy's lab. Farady (see Figure 4.1.1) was one of the
greatest experimenters ever. In the course of his experiments, he
discovered that a suspended magnet would revolve around a current
bearing-wire. This observation led him to propose that magnetism is a
circular force. He invented the dynamo in 1821, with which a large
amount of our current electricity is generated. In 1831, he discovered
electromagnetic induction. One of his most important contributions to
546
4.1 Introduction
physics in 1845 was his development of the concept of a field to describe
magnetic and electric forces.
Figure 4.1.1.
Michael Faraday: born September 22, 1791; died August 25, 1867.
Maxwell (see Figure 4.1.2) started out by writing a paper entitled "On
Faraday's Lines of Force" (1856), in which he translated Faraday's theories
into mathematical form. This description of Faraday's findings by means of
mathematics presented the lines of force as imaginary tubes containing an
incompressible fluid. In 1861, he published the paper "On Physical Lines
of Force" in which he treated the lines of force as real entities. Finally, in
1865, he published a purely mathematical theory known as "On a
Dynamical Theory of the Electromagnetic Field". The equations derived
by Maxwell and published in "A Treaties on Electricity and Magnetism"
(1873) are still valid and a source of basic laws for engineering as well as
physics.
4. Electrodynamics
Figure 4.1.2.
547
James Clerk Maxwell: born June 13, 1831; died November 5, 1879.
The aim of this chapter is to introduce basic phenomena and basic solution
procedures for electric fields. The material discussed is a collection of
examples. It is far from being complete by considering the huge diversity
of electromagnetic phenomena. However, the examples discussed
demonstrate how symbolic computations can be used to derive solutions
for electromagnetic problems.
This chapter is organized as follows: Section 4.2 contains material on
point charges. The exampl discuss the electric field of an assembly of
discrete charges distributed in space. In Section 4.3, a standard boundary
problem from electrostatics is examined to solve Poisson's equation for an
angular segment. The dynamical interaction of electric fields and charged
particles in a Penning trap is discussed in Section 4.4.
548
4.2 Discrete Charge Distributions
4.2 Potential and Electric Fields of Discrete
Charge Distributions
In electrostatic problems, we often need to determine the potential and the
electric fields for a certain charge distribution. The basic equation of
electrostatics is Gauss' law. From this fundamental relation connecting the
charge density with the electric field, the potential of the field can be
derived. We can state Gauss' law in differential form by
В?
(4.2.1)
div E = 4pr(r?).
В?
If we introduce the potential F by E = -grad F, we can rewrite Eq. (4.2.1)
for a given charge distribution r in the form of a Poisson equation
DF = - 4 pr
(4.2.2)
where r denotes the charge distribution. To obtain solutions of Eq.
(4.2..2), we can use the Green's function formalism to derive a particular
solution. The Green's function G(r?, ?r') itself has to satisfy a Poisson
equation where the continuous charge density is replaced by Dirac's delta
function Dr G Hr?, r? 'L = -4 p dHr? - r? 'L. The potential F is then given by
FHr?L = ?V GHr?, r? 'L rHr? 'L d 3 r'.
(4.2.3)
In addition, we assume that the boundary condition G ╩V = 0 is satisfied on
the surface of volume V . If the space in which our charges are located is
infinitely extended, the Green's function is given by
1
GIr?, r 'M =
1
ееееееее
еееее
╩ ?r-r? '╩
(4.2.4)
The solution of the Poisson equation (4.2.3) becomes
?
rHr 'L 3
еееее d r'.
FHr?L = ? ееееееее
╩ r?-r? '╩
(4.2.5)
Our aim is to examine the potential and the electric fields of a discrete
charge distribution. The charges are characterized by a strength qi and are
located at certain positions ?ri . The charge density of such a distribution is
given by
N
r(r?) = ?i=1
qi dH ?r i L.
(4.2.6)
4. Electrodynamics
549
The potential of such a discrete distribution of charges is in accordance
with Eq. (4.2.5):
N
F(r?) = ?
i=1
qi
ееееееее
еееее ,
╩ r?-r? ╩
(4.2.7)
i
where ?ri denotes the location of the point charge. The corresponding
electrical field is given by
?r
В?
N
qi ?r - ееееееее
еiеееееее
E Hr?L = -?i=1
?
╩ r-r? ╩ 3
i
(4.2.8)
and the energy density of the electric field of such a charge distribution is
given by
1 В? 2
ееее ? E ? .
w = ееее
8p
(4.2.9)
Three fundamental properties of a discrete charge distribution are defined
by Eqs. (4.2.7), (4.2.8), and (4.2.9). In the following, we write a
Mathematica package which computes the potential, the electric field, and
the energy density for a given charge distribution. With this package, we
are able to create pictures of the potential, the electric field, and the energy
density.
In order to design a graphical representation of the three quantities, we
need to create contour plots of a three-dimensional space. To simplify the
handling of the functions, we enter the cartesian coordinates of the
locations and the strength of the charges as input variables in a list.
Sublists of this list contain the information for specific charges. The
structure of the input list is given by 88x1 , y1 , z1 , r1 <, 8x2 , y2 , z2 , r2 <, ?<.
To make things simple in our examples, we choose the y = 0 section of the
three-dimensional space. The package PointCharge`, located in the
section on packages and programs, contains the equations discussed above.
The package generates contour plots of the potential, the electric field, and
the energy density.
In order to test the functions of this package, let us consider some
ensembles of charges frequently discussed in literature. Our first example
describes two particles carrying the opposite charge, known as a dipole.
Let us first define the charges and their coordinates by
550
4.2 Discrete Charge Distributions
charges = {{1,0,0,1},{-1,0,0,-1}}
881, 0, 0, 1<, 81, 0, 0, 1<<
The charges are located in space at x = 1, y = 0, z = 0 and at x = -1,
y = 0, z = 0. The fourth element in the sublists specifies the strength of the
charges. The picture of the contour lines of the potential is created by
calling
FieldPlot[charges,"Potential"];
1.5
1
0.5
0
-0.5
-1
-1.5
-1.5
Figure 4.2.3.
-1
-0.5
0
0.5
1
1.5
Contour plot of the potential for two charges in the Hx, zL-plane. The particles carry opposite
charges.
The second argument of FieldPlot[] is given as a string specifying the type
of the contour plot. Possible values are Potential, Field, and
EnergyDensity.
4. Electrodynamics
551
A graphical representation of the energy density follows by
FieldPlot[charges,"EnergyDensity"];
1.5
1
0.5
0
-0.5
-1
-1.5
-1.5
Figure 4.2.4.
-1
-0.5
0
0.5
1
Contour plot of the energy density of two charges in the Hx, zL-plane.
The electrical field of the two charges are generated by
1.5
552
4.2 Discrete Charge Distributions
FieldPlot@charges, "Field"D;
Since the generation of field plots is very flexible, we are able to examine
any configuration of charges in space. A second example is given by a
quadruple consisting of four charges arranged in a spatial configuration.
The locations and strength of the charges are defined by
quadrupole = 881, 0, 0, 1<,
81, 0, 0, 1<, 80, 0, 1, 1<, 80, 0, 1, 1<<
881, 0, 0, 1<, 81, 0, 0, 1<,
80, 0, 1, 1<, 80, 0, 1, 1<<
The potential is
4. Electrodynamics
553
FieldPlot@quadrupole, "Potential"D;
1.5
1
0.5
0
-0.5
-1
-1.5
-1.5
-1
-0.5
0
0.5
The field lines in the Hx, zL-plane with y = 0 are
1
1.5
554
4.2 Discrete Charge Distributions
FieldPlot@quadrupole, "Field"D;
The energy density looks like
4. Electrodynamics
555
FieldPlot@quadrupole, "EnergyDensity"D;
1.5
1
0.5
0
-0.5
-1
-1.5
-1.5
-1
-0.5
0
0.5
1
1.5
4.3 Boundary Problem of Electrostatics
In the previous section, we discussed the arrangement of discrete charges.
The problem was solved by means of the Poisson equation for the general
case. We derived the solution for the potential using
Df = 4 pr.
(4.3.10)
Equation (4.3.10) is reduced to the Laplace equation if no charges are
present in the space:
Df = 0.
(4.3.11)
The Laplace equation is a general type of equation applicable to many
different theories in physics, such as continuum theory, gravitation,
hydrodynamics, thermodynamics, and statistical physics. In this section,
we use both the Poisson and the Laplace equations (4.3.10) and (4.3.11) to
556
4.3 Boundary Problem
describe electrostatic phenomena. We show that Eqs. (4.3.10) and (4.3.11)
are solvable by use of Green's function. If we know the Green's function of
the equation, we are able to consider general boundary problems. A
boundary problem is defined as follows: For a certain volume V , the
surface of this volume, ≥ V , possesses a specific electric potential. The
problem is to determine the electric potential inside the volume given the
value on the surface. This type of electrostatic boundary problem is called
a Dirichlet boundary value problem. According to Eq. (4.3.10), there are
charges inside volume V . The distribution or density of these charges is
В?). The mathematical problem is to find solutions for Eq.
denoted by r(x
(4.3.10) or (4.3.11) once we know the distribution of charges and the
electric potential on the surface of the domain.
The Green's function allows us to simplify the solution of the problem. In
our problem, we have to solve the Poisson equation (4.3.10) under certain
restrictions. The Green's function related to the Poisson problem is defined
by
D GH Вx?, Вx? 'L = -4 pd H Вx? - Вx? 'L
(4.3.12)
under the specific boundary condition
В?, Вx? 'L ю
GHx
≥V
= 0
with Вx?' e ≥ V
(4.3.13)
on the surface ≥V of volume V .
In the previous section, we discussed the Green's function for an infinitely
extended space and found that the Green's function is represented by
GH Вx?, Вx? 'L = 1 Й ╩ Вx? - Вx? ' ╩. The present problem is more complicated than
the one previously discussed. We need to satisfy boundary conditions for a
finite domain in space.
For our discussion, we assume that the Green's function exists and that we
can use it to solve the boundary problem. The proof of this assumption is
given by Arfken [4.1]. The connection between the Green's function and
the solution of the boundary problem is derived using Gauss's theorem.
The first formula by Green
В? 3
В? 2 В?
(4.3.14)
?V div A d x =?V A d f ,
4. Electrodynamics
557
В?
along with an appropriate representation of the vector field A = F Ъ ?G ?F Ъ G yields the second formula by Green:
В?
(4.3.15)
div A = F Ъ D G - DF Ъ G.
Using the integral theorem of Gauss in the form of Eq. (4.3.14), we find
≥G
≥F
3
(4.3.16)
ееее - G ееее
еее M d 2 f ,
?V H F Ъ D G - DF Ъ GL d x = ?≥V IF ееее
≥n
≥n
where ≥ Й ≥ n = Вn? Ъ ? is the normal gradient. If we use relations (4.3.10),
(4.3.12), and (4.3.13) in Eq. (4.3.16), we can derive the potential by the
two integrals
В?L =
В? В?
В?
3
FHx
?V GHx , x 'L rHx 'L d x ' В?, Вx? 'L
≥GHx
1
В? 'L ееееееее
ееее
е
ее
е
FHx
е
еееееееееее d 2 f '.
?
4 p ≥V
≥n'
(4.3.17)
A comparison between Eqs. (4.3.17) and (4.2.3) reveals that the total
potential in the Dirichlet problem depends on a volume part (consistent
with Eq. (4.2.3) and on a surface part as well. The potential F at location Вx?
consists of a volume term containing the charges and of a surface term
В?L. The potential FHx
В? 'L used in the
determined by the electric potential FHx
surface term is known as a boundary condition. If there are no charges in
the present volume, solution (4.3.17) reduces to
В?, Вx? 'L
≥GHx
В?L = - ееее1ееее
В? 'L ееееееее
FHx
ееееееее
ееее d 2 f '.
FHx
4p ?
≥n'
≥V
(4.3.18)
For the charge-free case, the electric potential at a location Вx? inside the
В? 'L.
volume V is completely determined by the potential on the surface FHx
We are able to derive Eqs. (4.3.17) and (4.3.18) provided that the Green's
В?, Вx?') vanishes on the surface of V . In other words, we assume
function G(x
the surface potential to be a boundary condition. This type of boundary
condition is called a Dirichlet boundary condition. A second type is the
so-called von Neumann boundary condition, which specifies the normal
derivative of the electrostatic potential ≥ F Й ≥ n on the surface. A third type
used in potential theory is a mixture of Dirichlet and von Neumann
boundary conditions. In the following, we will restrict ourselves to
Dirichlet boundary conditions only.
If we take a closer look at solutions (4.3.17) and (4.3.18) of our boundary
value problem, we observe that the Green's function as an unknown
558
4.3 Boundary Problem
determines the solution of our problem. In other words, we solved the
boundary problem in a form which contains an unknown function as
defined by relation (4.3.12) and the boundary condition (4.3.13). The
central problem is to find an explicit representation of the Green's
function. One way to tackle this is by introducing an eigenfunction
expansion [4.2]. This procedure always applies if the coordinates are
separable. The eigenfunction expansion of the Green's function is based on
the analogy between an eigenvalue problem and equations (4.3.10) and
(4.3.11) for the potential.
The eigenvalue problem related to equation (4.3.10) is given by
Dy+(4p r +l)y = 0.
(4.3.19)
For a detailed discussion of the connection, see [4.2]. We assume that
solutions y of Eq. (4.3.19) satisfy the Dirichlet boundary conditions. In
this case, the regular solutions of Eq. (4.3.12) only occur if parameter l =
ln assumes certain discrete values. The ln 's are the eigenvalues of Eq.
(4.3.19). Their corresponding functions yn are eigenfunctions. The
eigenfunctions yn are orthogonal and satisfy
В? 3
* В?
?V ym Hx L yn Hx L d x = dmn .
(4.3.20)
The eigenvalues of Eq. (4.3.19) can be discrete or continuous. In analogy
to Eq. (4.3.12), the Green's function has to satisfy the equation
В?, Вx? 'L + H4 p r + lL GHx
В?, Вx? 'L = - 4 p dHx
В? - Вx? 'L,
Dx GHx
(4.3.21)
where l is different to the eigenvalues ln . An expansion of the Green's
function with respect to the eigenfunctions of the related eigenvalue
problem is possible if the Green's function satisfies the same boundary
conditions. Substituting an expansion of the Green's function
В?, Вx? 'L = ? a Hx
В? 'L y Hx
В?L
GHx
n
n n
(4.3.22)
into Eq. (4.3.21), we get
В? 'L Hl - l L y Hx
В?
В? В?
(4.3.23)
?m am Hx
m
m L = -4 p dHx - x 'L.
Multiplying both sides of Eq. (4.3.23) by y*n H Вx? L and integrating the result
over the entire volume, we obtain the expansion coefficients am H Вx? 'L.
Using the orthogonal relation (4.3.20) simplifies the sum. The expansion
coefficients are defined by
4. Electrodynamics
559
В? 'L
y*n Hx
В? 'L = 4 p ееееееее
an Hx
ееееее .
ln -l
(4.3.24)
With relation (4.3.24) we get the representation of the Green's function
В?'L y Hx
В?L
y* Hx
В?, Вx? 'L = 4 p
GHx
?
n
n
ееееееееееееееее
ln -lеееееееее .
n
(4.3.25)
So far, our considerations have assumed a discrete spectrum of
eigenvalues. For a continuous distribution of eigenvalues ln , we need to
replace the sum in Eq. (4.3.25) with an integral over the eigenvalues.
By using the representation of the Green's function (4.3.25), we can
rewrite the solution of the potential (4.3.17) and (4.3.18) in the form
В?L =
FHx
? 4p?
V
?
≥V
n
В?'L y Hx
В?
y*n Hx
n L
ееееееееееееееее
ееее
еееее rHxВ? 'L d 3 x' ln -l
В? 'L
FHx
?
n
В?
В?'L
≥y* Hx
yn HxL
n
ееееееее
ееее ееееееее
ееееееее d 2 f '
ln -l
≥n'
В?L
yn Hx
ееее ?V y*n HxВ? 'L rHxВ? 'L d 3 x' = 4 p ? ееееееее
l
-l
n
n
В?'L
В?L
≥y* Hx
y Hx
?
n
(4.3.26)
n
n
ееееееее
ееее ? FHxВ? 'L ееееееее
ееееееее d 2 f '.
ln -l
≥n'
≥V
If we know the eigenfunctions and eigenvalues of the problem, we can
represent the potential by
В?L = ? Hc - d L y Hx
В?L,
FHx
n
n
n
n
(4.3.27)
where the cn 's and the dn 's are expansion coefficients defined by
4p
ееее y* HxВ? 'L rHxВ? 'L d 3 x'
cn = ееееееее
ln -l ?V n
(4.3.28)
В?'L
≥y*n Hx
1
В? 'L ееееееее
е
ее
е
FHx
е
еее
ееее d 2 f '.
dn = ееееееее
?
ln -l
≥n'
≥V
(4.3.29)
and
For the charge-free case r = 0, we find
В?L = FHx
?
В?'L
≥y* Hx
В?
n
yn HxL
В? 'L ееееееееnееееееее d 2 f '.
ееееееее
ееее
FHx
ln -l ?
≥n'
(4.3.30)
≥V
which reduces to
В?L = - ? d y Hx
В?L .
FHx
n n n
(4.3.31)
The unknown quantities of this representation are the eigenfunctionsyn
and the expansion coefficients cn and dn . By examining a specific planar
560
4.3 Boundary Problem
problem, we show how these unknowns are calculated. To make things
simple, we assume that no charges are distributed on the plane.
The problem under consideration examines in a section of a disk in which
boundaries have fixed potential values FHr, j = 0L = 0, FHr, j = aL = 0,
and FHr = R, jL = F0 Hj L. The specific form of the domain and the
boundary values are given in Figure 4.3.5.
FHr,j=aL=0
FHr=R,jL=F0
G
a
Figure 4.3.5.
R
FHr,j=0L=0
Boundary conditions on a disk segment. The domain G is free of charges.
The domain G is free of any charges and the potential FHr, jL is regular
and finite for r ь 0. To solve the problem efficiently, we choose
coordinates which reflect the geometry of our problem. In this case, they
are plane cylindrical coordinates. Since G is free of any charges, Laplace's
equation in plane cylindrical coordinates takes the form
2
≥
≥F
1 ≥ F
ееее1r ееее
ее Ir ееее
еее M + ееее
е еееееееее = 0.
≥r
≥r
r2 ≥j2
(4.3.32)
When deriving the solution, we assume that the coordinates are separated.
If we use the assumption of separating the coordinates, we are able to
express the electric potential as FHr, jL = gHrL hHj L. Substituting this
expression into Eq. (4.3.32), we get
2
dg
d hHjL
d
r
1
2
ееее
еееее ееее
еееее ееееееее
2 ееее = n ,
gHrL
d еrее Ir ееее
d еrеее M = - ееее
hHjL
d jееее
(4.3.33)
4. Electrodynamics
561
where n is a constant. Separating both equations, we get two ordinary
differential equations determining g and h. g and h represent the
eigenfunctions of the Green's function
dg
d
r
2
ееее
еееее ееее
gHrL
d еrее Ir ееее
d еrеее M = n ,
(4.3.34)
d 2 hHjL
1
ееее
ееееее ееееееее
ееееееее = -n2 ,
hHjL
d j2
(4.3.35)
The eigenfunctions of the radial part of the potential are
gn HrL = an rn + bn r-n .
(4.3.36)
The angular part of the eigenfunctions defined in Eq. (4.3.35) is given by
hn HjL = An sin HnjL + Bn cosHnj L.
(4.3.37)
The solutions (4.3.36) and (4.3.37) contain four constants an , bn , An , and
Bn for each eigenvalue n. These constants have to satisfy the boundary
conditions and the condition of regularity at r = 0.
Let us first examine the radial part of the solution in the domain G. We
find that for j = 0, the relation
FHr, j = 0L = gHrL hHj = 0L = 0
(4.3.38)
needs to be satisfied. From condition (4.3.38), it follows that
hHj = 0L = Bn = 0. From the boundary condition at j=a we get the
condition
FHr, j = aL = gHrL hHj = aL = 0,
(4.3.39)
which results in hHaL = An sinHnaL = 0. As a consequence, we get n =
n p Й a with n = 0, 1, 2, 3, ... . The angular part of the solution thus reduces
to
np
hn HjL = An sin H ееееaееее jL.
(4.3.40)
From the condition of regularity FHr ь 0, jL < ╤, it follows from
FHr, j L = hn HjL Han rn + bn r-n L
(4.3.41)
that bn = 0. The solution of the potential is thus represented by
np
n pЙa
FHr, jL = ?╤
sin H ееееaееее jL,
n=0 dn r
(4.3.42)
where dn = an An . Expression (4.3.42) contains the unknown coefficients
dn , which we need to determine in order to find their explicit
562
4.3 Boundary Problem
representations. Values for dn are determined by applying the boundary
condition on the circle FHr = R, jL = F0 HjL. If we take into account the
orthogonality relation for the trigonometric functions
a
np
mp
ееееa2е ?0 sinH ееее
ееее jL sinH ееее
еееее jL d j = dmn ,
a
a
(4.3.43)
we are able to derive from the boundary condition of the circle a
representation of dn by
a
np
?0 F0 HjL sinH ееееaееее jL d j =
╤
a
np
mp
?m=0 dm Rm pЙa ?0 sinH ееееaееее jL sinH ееееaеееее jL d j
╤
a
= ?
dm Rm pЙa ееее
еd
2 nm
m=0
a n pЙa
= ееее
е R dn ,
2
(4.3.44)
or in explicit form,
2
a
np
dn = R-n pЙa ееее
е F HjL sinH ееееaееее jL d j.
a ?0 0
(4.3.45)
The representation of dn by the integral (4.3.45) includes the boundary
condition and only contains known parameters. Thus, we can determine
dn 's numerical value if we know the boundary condition and if we specify
the index m of the expansion in Eq. (4.3.42). The values of dn are,
however, only defined if the integral in Eq. (4.2.45) converges. The
specific form of the Green's function is derivable if we compare the
representation of the solution (4.3.42) with the definition of the Green's
function.
4. Electrodynamics
563
With the above theoretical considerations, an explicit representation of the
solution is now necessary. By specifying the geometrical parameters of the
problem, the radius R of the segment, the angle a, the potential value along
the rim of the disk and Eq. (4.3.42), we can calculate the potential in the
domain G. The central quantities of the expansion (4.3.42) are the
coefficients dn . In order to make these factors available, we define the sum
(4.3.42) and the integral (4.3.45) in the Potential[] function of the package
BoundaryProblem` (see Section 4.6.2 for details). We define relations
(4.3.42) and (4.3.45) to control the accuracy of the calculation using an
upper summation index n (see also the definition of the function
Potential[] in Section 4.6.2). An example of the potential for the
parameters R=1, a = p Й 4 and F0 (j)=1 is given in Figure 4.3.6. The calling
sequence of Potential[] takes the form Potential@ f @xD, R, a, nD.
S
PotentialA1, 1, ccccc , 10E;
4
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
564
4.3 Boundary Problem
Contour plot of the potential in the domain G. Boundary conditions and geometric
parameters are F0 (j)= 1, R=1, a = p Й 4 and n=10.
Figure 4.3.6.
The result shows an approximation of the potential up to order 10. The
contour lines show that the approximation shows some wiggles at the rim
of the domain. The quality of the approximation can be checked by
increasing the approximation order. The increase in quality is shown in the
following sequence of plots (Figure 4.3.7):
S
pl = TableAPotentialA1, 1, ccccc , iE, 8i, 1, 20, 2<E;
4
1
0.8
0.6
0.4
0.2
0
0
Figure 4.3.7.
0.2
0.4
0.6
0.8
1
Sequence of contour plot of the potential in the domain G. Boundary conditions and
geometric parameters are F0 (j)= 1, R=1, a = p Й 4 and ne[1,20,2].
At this place, a word of caution should be mentioned. The approximation
of the potential shows that the procedure is sensitive in the approximation
order. The kind of calculation is also sensitive on the boundary conditions,
which is given as first argument in the function Potential[]. Although the
calculated potential shows the expected behavior, it is not always possible
4. Electrodynamics
565
to calculate the potential for a reasonable approximation order for arbitrary
boundary conditions. This shortcoming is due to the calculation of
integrals in the procedure. However, the reader should experiment with the
function and test the limitations of the method to gain a feeling for the
applicability. An example with a spatially varying boundary condition on
the rim is presented in Figure 4.3.8.
S
PotentialA2 + Sin@7 ID, 1, ccccc , 20E;
4
1
0.8
0.6
0.4
0.2
0
0
Figure 4.3.8.
0.2
0.4
0.6
0.8
1
Contour plot of the potential in the domain G. Boundary conditions and geometric
parameters are F0 (j)= 2+sin(7j), R=1, a = p Й 4 and n=10.
566
4.4 Penning Trap
4.4 Two Ions in the Penning Trap
The study of spectroscopic properties of single ions requires that one or
two ions are trapped in a cavity. Nowadays, ions can be successfully
separated and stored by means of ion traps. Two techniques are used for
trapping ions. The first method uses a dynamic electric field, while the
second method uses static electric and magnetic fields. The dynamic trap
was originally invented by Paul [4.3]. The static trap is based on the work
of Penning [4.4]. Both traps use a combination of electric and magnetic
fields to confine ions in a certain volume in space. Two paraboloids
connected to a dc-source determine the kind of electric field in which the
ions are trapped. The form of the paraboloids in turn determines the field
of the trap's interior. Since the motion of the ions in Paul's trap is very
complicated, we restrict our study to the Penning trap.
In our discussion of the Penning trap, the form of the quadrupole fields
determined by the shapes of the paraboloids is assumed to be
U
е0еееееее Hx2 + y2 - 2 z2 L,
F = ееееееее
r20 +2 z20
(4.4.46)
where U0 is the strength of the source and r0 and z0 are the radial and axial
extensions of the trap (see Figure 4.4.9). The shape of the potential is a
consequence of the Laplace equation DF=0. The given functional shape of
the potential is experimentally created by conducting walls which are
connected to a dc-battery. The force acting on an ion carrying charge q in
the trap is given by
В?
В?
(4.4.47)
F = q E= -q ?F.
4. Electrodynamics
567
-2
2
2
1
0
-1
0
1
2
-1
-2
2
0
-2
Figure 4.4.9.
Cross-section of the Penning trap. The paraboloids are positioned on dc-potentials. A
constant magnetic field is superimposed in the z vertical direction (not shown). The ions
move in the center of the trap.
В?
From the functional form of the electric field E of the trap
ij x yz
jj
z
2 U0
j y zzz = - ееееееее
ееееееее HxВ? - 3 e?z L,
zz
r0 +2 z0 jjj
r20 +2 z20
k -2 z {
В?
2 U0
E = -?F = - ееееееее
2 ееееееее
2
(4.4.48)
we detect a change of sign in the coordinates. This instability allows the
ions to escape the trap. To prevent escape from the trap in the z-direction,
Paul and co-workers used a high-frequency ac-field and Penning and
В?
co-workers used a permanent magnetic field B = B0 e?z .
In a static trap the forces acting on each of the two ions are determined by
the electromagnetic force of the external fields and the repulsive force of
the Coulomb interaction of the charges. The external fields consist of the
static magnetic field along the z-axis and the electric quadrupole field of
the trap. The Coulomb interaction of the two particles is mainly governed
568
4.4 Penning Trap
by the charges which are carried by the particles. The total force on each
particle is a combination of trap and Coulomb forces. Since we have a
system containing only a few particles, we can use Newton's theory (see
section 2.4) to write down the equations of motion in the form
В ? T В ? Coul
m Вx? '' = HF Li + HF Li
(4.4.49)
i=1,2.
В? T
HF Li
In equation (4.4.49) the trap force
denotes the Lorentz force of a
particle in the electromagnetic field given by
В? T
В?
В?
(4.4.50)
H F Li = q HELi + q Iv?i ╣ BM.
В?
Since the magnetic field B is a constant field along the z-direction
В?
(4.4.51)
B= B0 e? z ,
the total trap force on the ith ion is given by
В? T
В?
2 U0
ееееееее HxВ? - 3 zi e?z L + q IxВ? 'i ╣ BM .
HF Li = - ееееееее
r20 +2 z20
(4.4.52)
The Coulomb forces between the first and the second ion are
Вx? -x
В?
В ? Coul
q2
1
2
ееее ееееееее
ее ,
HF L12 = ееееееее
? ееееееее
В
4 p╤0 ВВВ
╩ x1 -x?2 ╩3
В
?
В
?
2
В ? Coul
q
x2 -x 1
HF L21 = ееееееее
ееее ееееееее
ВВВ? ееееееее
В? е3е .
4 p╤
0
(4.4.53)
(4.4.54)
╩ x1 -x2 ╩
The explicit forms of the equations of motion are thus
m Вx? ''1 =
Вx? -x
В?
В?
q2
2 U0
В? - 3 z e? L + q Ix
В? ' ╣ B
1
2
е
еее
е
еее
Hx
е
еее
ееееееее
е
еееееее
ее ,
- ееееееее
M + ееееееее
ВВВ
?
1
1
z
1
2
2
В
4 p╤0 ╩ x1 -x?2 ?3
r0 +2 z0
2 U0
В? - 3 z e? L +
m Вx? ''2 = - ееееееееееееееее
еееееееее Hx
2
2 z
2
r0 + 2 z20
Вx? - Вx?
q2
В?
2
1
В? ' ╣ B
ееееееее
ееееееее3е .
q Ix
M + еееееееееееееееее ееееееееееееееее
2
В
?
В
?
4 p╤ ╩ x - x ╩
0
1
(4.4.55)
(4.4.56)
2
The two equations of motion (4.4.55) and (4.4.56) are coupled ordinary
differential equations of the second order. They can be decoupled by
introducing relative and center of mass coordinates:
?r = Вx? - Вx? ,
2
В? 1
R = ееее12 H Вx?1 + Вx?2 L.
(4.4.57)
4. Electrodynamics
569
Using Eqs. (4.4.57) in (4.4.55) and (4.4.56), we can describe the motion of
the two ions in the center of mass and in relative coordinates. The two
transformed equations read
В?
В?
q B0 В? ?
2 U0
ееееее IR - 3 Z e?z M + ееее
ееееее IR ' ╣ ez M,
R '' = - ееееееееееееееее
m
mHr20 +2 z20 L
2 U0
r? '' = - ееееееееееееееее
ееееееееееееееее
ееее Hr? - 3 z e? z L +
2
2
mHr0 + 2 z0 L
?r
q2
q B0
ееееееееееееее Hr? ' ╣ e?z L + еееееееееееееееееееееееее ееееееее
еееееее .
?
m
2 p m ╤0 ╩ r ╩3
(4.4.58)
(4.4.59)
If we assume that the two ions carry a negative charge q < 0 and that the
dc-potential U0 on the paraboloids is positive HU0 > 0L, then we can
introduce two characteristic frequencies and a scaled charge by
2U
0
ееееее ,
w20 = ееееееееееееееее
mHr20 +2 z20 L
╩q╩ B0
ееее ,
wc = ееееееее
m
(4.4.60)
Q2 = ееееееее
еееееееее .
2 p m ╤0
(4.4.62)
(4.4.61)
q2
Constant w0 denotes the frequency of the oscillations along the z-direction.
wc is the cyclotron frequency (i.e., the frequency with which the ions spin
around the magnetic field). Q represents the scaled charge. Using these
constants in the equations of motion (4.4.58) and (4.4.59), we get a
simplified system of equations containing only three constants:
В?
В?
В?
R '' = w20 IR - 3 Z e? z M - wc IR ' ╣ e?z M,
(4.4.63)
?r '' = w2 Hr? - 3 z e? L - w Hr? ' ╣ e? L + Q2 ееее?rеееее .
(4.4.64)
0
z
c
z
╩ Вr?╩3
In the following subsections, we discuss the two different types of motion
resulting from these equations.
4.4.1 The Center of Mass Motion
The center of mass motion is determined by Eq. (4.4.63). Writing down
the equations of motion in cartesian coordinates X , Y, and Z, we get a
coupled system of equations:
X '' - w20 X + wc Y '= 0,
Y '' - w20 Y - wc X '= 0,
Z '' + 2 w20 Z= 0.
(4.4.65)
(4.4.66)
(4.4.67)
570
4.4 Penning Trap
The equations of motion for the X - and Y- components are coupled
through the cross-product. The Z- component of the motion is completely
decoupled from the X and Y coordinates. The last of these three equations
Х!!!!
is equivalent to a harmonic oscillator with frequency 2 w0 . Thus, we
immediately know the solution of the Z- coordinate given by
Х!!!!
ZHtL = A cosI 2 w0 t + BM.
(4.4.68)
The arbitrary constants A and B are related to the initial conditions of the
motion by ZHt = 0L = Z0 and Z ' Ht = 0L=Z0' . Therefore, A = Z02 + Z '20 Й 2 w20
Х!!!!
and tan B = Z0' К 2 w0 Z0 .
A representation of the solution of the remaining two equations (4.4.65)
and (4.4.66) follows if we combine the two coordinates X and Y by a
complex transformation of the form @ = X + i Y . Applying this
transformation to the two equations delivers the simple representation
?
╟
(4.4.69)
@ - w20 @ - i wc @ = 0.
If we assume that the solutions of Eq. (4.4.69) are harmonic functions of
the type @ = ei w t , we get the corresponding characteristic polynomial
wHwc - wL - w20 = 0.
(4.4.70)
The two solutions of this quadratic equation are given by the frequencies
w1 and w2 :
w 2
wc
еее + $%%%%%%%%%%%%%%%%%%%%%%%%%
H ееее2еcе L - w20% ,
w1 = ееее
2
2
wc
w2 = ееее
еее - $%%%%%%%%%%%%%%%%%%%%%%%%%
H ееее2еcе L - w20% .
2
w
(4.4.71)
(4.4.72)
The two frequencies are combinations of the cyclotron frequency wc and
the axial frequency w0 . The general solution of Eqs. (4.4.65) and (4.4.66)
is thus given by
X HtL =
Br cosHw1 tL + Bi sinHw1 tL + Ar cosHw2 tL + Ai sinHw2 tL,
(4.4.73)
Y HtL =
Ar sinHw2 tL - Ai cos Hw2 tL + Br sinHw1 tL - B cosHw1 tL.
(4.4.74)
4. Electrodynamics
571
The constants of integration Ar , Ai , Br , and Bi are related to the initial
conditions X0 , Y0 , X0' , and Y0' by the relations
'
Ar =
Y0 -w1 X0
ееееееееееееееее
w2 -w1еееее ,
X0' +w1 Y0
еееее ,
Ai = ееееееееееееееее
w2 -w1
Y0' -w2 X0
еееее ,
Br = ееееееееееееееее
w1 -w2
X0' +w2 Y0
Bi = ееееееееееееееее
w1 -w2еееее .
(4.4.75)
(4.4.76)
(4.4.77)
(4.4.78)
A special case of solutions (4.4.73) and (4.4.74) is obtained if we assume
that the center of mass is initially located in the origin of the coordinate
system X0 = Y0 = 0. We get from (4.4.75) Ar = -Br , and Ai = -Bi . The
solution then takes the form
ji
zy
w
w 2
H ееее2еcе L - w20% tzzzz X HtL = Ar sinH ееее2еcе tL sinjjjj$%%%%%%%%%%%%%%%%%%%%%%%%%
k
{
yz
ji
w
w 2
H ееее2еcе L - w20% tzzzz,
Ai cos H ееее2еcе tL sinjjj$%%%%%%%%%%%%%%%%%%%%%%%%%
j
k
{
i
j
zy
2
w
w
H ееее2еcе L - w20% tzzzz Y HtL = Ai sinH ееее2еcе tL sinjjjj$%%%%%%%%%%%%%%%%%%%%%%%%%
k
{
yz
ji
w
w 2
H ееее2еcе L - w20% tzzzz.
Ar cos H ееее2еcе tL sinjjjj$%%%%%%%%%%%%%%%%%%%%%%%%%
k
{
(4.4.79)
(4.4.80)
The above solutions show that the motion of the center of mass in the
HX , YL-plane is governed by two frequencies. The first frequency is
one-half of the cyclotron frequency wc and the second frequency is a
combination of the axial frequency and the cyclotron frequency given by
"############################
#
Hwc Й 2L2 - w20 . A plot of the motion in center of mass coordinates is
given in Figure 4.4.10. The three-dimensional motion of the center of mass
Х!!!!
is governed by three frequencies. The axial frequency 2 w0 determines
the oscillation rate of the center of mass along the z-axis. The halved
cyclotron frequency wc Й 2 governs the spinning of the particles around the
magnetic lines.
572
4.4 Penning Trap
0.2
2
z
0
0.5
-0.2
.2
0
-0.5
y
0
-0.5
x
0.5
Figure 4.4.10.
Motion of the center of mass in space for t ? @0, 100D. The initial conditions are
╟
╟
X0 = 0.5 = Y0 , X 0 = 0.1 = Y 0 . The cyclotron frequency is fixed at wc =5.
4.4.2 Relative Motion of the Ions
The relative motion of the two ions is governed by Eq. (4.4.64)
ВВВ? L - w Hr? ' ╣ e? L + Q2 ееее?rеееее .
?r '' = w2 Hr? - 3 ze
z
c
z
0
╩ r?╩3
(4.4.81)
Cylindrical coordinates are the appropriate coordinate system giving an
efficient description of the relative motion of the particles. Location ?r of
the relative particle is given in cylindrical coordinates by the representation
?r = r e? + z e? ,
r
z
(4.4.82)
where e? r and e?z represent the unit vectors in the radial and axial directions,
respectively.
Using these coordinates in the equation of motion (4.4.81) gives the
following representation:
Hr'' - r j'2 L e? r + H2 r' j' + rj''L e?j + z '' e?z -
Q2 Hr e? r +z e? z L
ееееееее
w20 Hr e? r - 2 z e? z L + wc H-r' e?j + rj' e? r L = ееееееееееееееее
3еееее .
Х!!!!!!!!!!!!!!!
2
2
I r +z M
(4.4.83)
4. Electrodynamics
573
Separating each coordinate direction, we can split Eq. (4.4.83) into a
system of equations for the coordinates r, j, and z:
Q2 r
ееее
ееее3 ,
r'' - r j'2 - w20 r + wc r j' = ееееееееееееееее
Х!!!!!!!!!!!!!!!
2
2
(4.4.84)
2 r' j' + r j ' - wc r = 0,
(4.4.85)
I r +z M
z '' + 2 w20
Q2 z
z = ееееееееееееееее
ееееееее3 .
Х!!!!!!!!!!!!!!!
I r2 +z 2 M
(4.4.86)
By multiplying Eq. (4.4.85) by the radial coordinate r and integrating the
result, we are able to derive an integral of motion. This integral of motion
is given by an extended angular momentum containing the cyclotron
frequency and is thus connected with the magnetic field. The conserved
quantity is given by
w
bB = r2 j' - ееее2еcе r2 .
(4.4.87)
The integral of motion (4.4.87) eliminates the j dependence in Eq.
(4.4.84). The elimination of j reduces the system of equations (4.4.84) and
(4.4.86) to
b2
2
w
Q2 r
B
ее = ееееееееееееееее
ееееееее3 ,
r'' + IH ееее2еcе L - w20 M r - ееее
Х!!!!!!!!!!!!!!!
r3
I r2 +z 2 M
Q2 z
z '' + 2 w20 z = ееееееееееееееее
ееее
ееее3 .
Х!!!!!!!!!!!!!!!
2
2
I r +z M
(4.4.88)
(4.4.89)
This system of equations contains a multitude of parameters. Our aim is to
reduce these parameters by appropriately scaling the temporal and spatial
coordinates. If we consider the expression b = Hwc Й 2L2 - w20 > 0 to be
positive, time is scaled by t = b t. The radial and axial coordinates r and z
2
are scaled by the factor d = HQ Й bL ее3ее . Introducing the abbreviations
2
Х!!!!
n2 = HbB Й bL2 and l2 = I 2 w20 Й bM simplifies the system of equations
(4.4.88) and (4.4.89) to
2
r
n
r'' + r - ееее
ее = ееееееееееееееее
ееее
ееее3 ,
Х!!!!!!!!!!!!!!!
r3
2
2
I r +z M
z
2
ееееееее3 ,
z '' + l z = ееееееееееееееее
Х!!!!!!!!!!!!!!!
I r2 +z 2 M
(4.4.90)
(4.4.91)
containing only two parameters n and l. The handling of Eqs. (4.4.90) and
(4.4.91) is easier than the four parameter representation in equations
(4.4.88) and (4.4.89). Note that Eqs. (4.4.90) and (4.4.91) are equivalent
574
4.4 Penning Trap
to the secular equations of the Paul trap. Both systems of equations are
derived from a Lagrangian given by
1
1
n2
1
Hr'2 + z '2 L - J ееее
Hr2 + 2 l2 z 2 L + ееееееее
еееееееееее + ееее
еееее N.
3 = ееее
Х!!!!!!!!!!!!!!!
2
2
2 r2
r2 +z 2
(4.4.92)
Equations (4.4.90) and (4.4.91) form a highly nonlinear coupled system of
equations which can only be solved analytically given a special choice of
parameters l and n [4.5]. If we wish to choose parameters, we need to
integrate the equations numerically. Mathematica supports numerical
integrations and we use this property to find numerical solutions for Eqs.
(4.4.90) and (4.4.91). The package Penning`, a listing is given in Section
4.6.3, contains the necessary function PenningI[] to integrate Eqs. (4.4.90)
and (4.4.91). Function PenningI[] also provides a graphical
representations of the potential and the path of the relative particle. An
example of a typical path in the potential is given in Figure 4.4.11.
Parameters l and n of this figure have been chosen so that the motion of
the relative particle is regular. Figure 4.4.12 shows a path for parameters l
and n where chaotic motion is present.
2.5
5
V 2
1.5
.5
2
1
0
-2
z
-1
-1
0
r
1
2
Figure 4.4.11.
-2
Relative motion in a Penning trap for l = 1 and n = 0. The plot of the particle is
superimposed on the effective potential. The numerical integration extends over
t ? @0, 100D. The initial conditions are r0 = 1.1, z0 = 0.5, r╟0 = 0.0, and E = 2.0.
4. Electrodynamics
575
6
2
V 4
2
1
0
0
-2
z
-1
-1
0
r
1
2
Figure 4.4.12.
-2
Relative motion in a Penning trap for l = 1.75 and n = 0. The plot of the particle is
superimposed on the effective potential. The numerical integration extends over
t ? @0, 100D. Initial conditions are r0 = 1.0, z0 = 0.0, r╟0 = 0.0, and E = 3.0.
Figures for different initial conditions and parameters can be generated for
example by
576
4.4 Penning Trap
PenningI@1.0, 0, 3, 0, 1.1, 100D;
4
V 3
2
2
1
0 z
-2
2
-1
0
r
-1
1
-2
2
1
PenningIA1.0, 0.1, 3.6, 0, cccccccccc , 100E;
Х!!!!
2
4
V
2
2
0
-2
2
0 z
-1
-2
0
r
1
2
4. Electrodynamics
577
The center of mass motion is accessible by the function PenningCMPlot[]:
PenningCMPlot@0.1, 0.2, 0.01, 0.01, 2.1D;
0.2
z 0
-0.2
0.5
0 y
-0.5
0
x
-0.5
0.5
0
5
4.5 Exercises
1. Create some pictures for a quadrupole arrangement of charges using
the package PointCharge'. Choose the location of the charges in the
representation plane of the potential section. What changes are
required if your choice of coordinates for the charges is outside the
representation plane? Perform some experiments with a larger number
of charges.
2. Examine the electric potential of a disk segment under several
boundary conditions using the package BoundaryProblem' (e.g., F0 =
sin(j) or F0 = j). What changes occur in the potential if we change the
angle a? Examine the influence of the upper summation index N on
the accuracy of the solution.
3. Study the dynamic properties of two ions in a Penning trap for the
following:
578
4.5 Exercises
a) A vanishing angular momentum (n=0) and different frequency ratios
l. Which l values result in chaotic motion and in a regular motion of
the particles?
b) Find solutions for n ° 0, l = 1 and l = 2.
c) Examine the parameter combination n = 0 and l = ееее12 .
4. Develop a Mathematica function to combine the relative and center
of mass coordinates for a representation of motion in real space for the
two-ion problem of a Penning trap.
5. Reexamine the Green's function formalism and discuss the problem
of a rectangular boundary with one side carrying a constant charge
distribution. The three other sides are fixed to the ground potential.
6. Examine a collection of three particles in a Penning trap.
7. Discuss the motion of two particles in a Penning trap for n° 0 and l
arbitrary.
4.6 Packages and Programs
4.6.1 Point Charges
Package for the generation of fields, potentials and energy densities.
BeginPackage["PointCharge`"];
(* --- load additional standard packages --- *)
Needs["Graphics`PlotField`"];
Clear[Potential,Field,EnergyDensity,FieldPlot];
(* --- export functions --- *)
Potential::usage = "Potential[coordinates_List]
creates the potential of
an assembly of point charges. The cartesian
coordinates of the locations of
the charges are given in the form of
4. Electrodynamics
579
{{x,y,z,charge},{x,y,z,charge},...}.";
Field::usage = "Field[coordinates_List] calculates
the electric field for
an ensemble of point charges. The cartesian
coordinates are
lists in the form of {{x,y,z,charge},{...},...}.";
EnergyDensity::usage =
"EnergyDensity[coordinates_List] calculates the
density of the energy for an ensemble of point
charges. The cartesian
coordinates are lists in the form of
{{x,y,z,charge},{...},...}.";
FieldPlot::usage =
"FieldPlot[coordinates_List,typ_,options___] creates
a
ContourPlot for an ensemble of point charges. The
plot type (Potential,
Field, or Density) is specified as string in the
second input variable. The
third argument allows a change of the Options of
ContourPlot and
PlotGradientField.";
(* --- define the global variables x,y,z
--- *)
x::usage;
y::usage;
z::usage;
Begin["`Private`"];
(* --- determine the potential --- *)
Potential[coordinates_List]:=
Block[{x,y,z},
Fold[Plus,0,Map[(#[[4]]/Sqrt[(x-#[[1]])^2 +
(y-#[[2]])^2 +
(z-#[[3]])^2])&, coordinates]]];
(* --- calculate the field ---*)
Field[coordinates_List]:=
Block[{field,x,y,z},
field = Fold[Plus,0,Map[(#[[4]]*({x,y,z}-Take[#,3])/
580
4.6 Packages and Programs
(Sqrt[(x-#[[1]])^2 +
(y-#[[2]])^2 +
(z-#[[3]])^2
])^3)&,coordinates]];
Simplify[field]
];
(* --- calculate the energy --- *)
EnergyDensity[coordinates_List]:=
Block[{density,x,y,z,field},
field = Field[coordinates];
density = field.field/(8*Pi)
];
(* --- create plots
--- *)
FieldPlot[coordinates_List,typ_,options___]:=
Block[
{pot, ncharges, xmin, xmax, zmin, zmax, xcoord
= {}, zcoord = {},
pl1, pl2},
ncharges = Length[coordinates];
(* --- determine limits for the plot --- *)
Do[
AppendTo[xcoord,coordinates[[i,1]]];
AppendTo[zcoord,coordinates[[i,3]]],
{i,1,ncharges}];
xmax = Max[xcoord]*1.5;
zmax = Max[zcoord]*1.5;
xmax = Max[{xmax,zmax}];
zmax = xmax;
xmin = -xmax;
zmin = xmin;
Clear[xcoord,zcoord];
(* --- fix the type of the plot ---*)
If[typ == "Potential",pot =
Potential[coordinates] /. y -> 0,
If[typ == "Field",pot =
-Potential[coordinates] /. y -> 0,
If[typ == "EnergyDensity",pot =
EnergyDensity[coordinates] /. y -> 0,
Print[" "];
Print[" wrong key word! Choose "];
Print[" Potential, Field or EnergyDensity
Print[" to create a plot "];
Return[]
"];
4. Electrodynamics
581
]]];
(* --- plot the pictures --- *)
If[typ == "Field",
pl1 =
PlotGradientField[pot,{x,xmin,xmax},{z,zmin,zmax},
options,
PlotPoints->20,
ColorFunction->Hue
],
pl1=
ContourPlot[pot,{x,xmin,xmax},{z,zmin,zmax},
options,
PlotPoints->50,
ColorFunction->Hue,
Contours->15]
]
];
End[];
EndPackage[];
4.6.2 Boundary Problem
The following package contains the main calculation steps for determining
the expansion coefficients in the harmonic series representation of the
potential.
BeginPackage@"BoundaryProblem`",
8"Calculus`Integration`"<D;
Clear@PotentialD;
Potential::usage =
"Potential@boundary_,R_,alpha_,n_D calculates the
potential in a circular segment. Input
parameters are the potential on the
circle, the radius R of the circle and the
angle of the segment of the circle.
The last argument n determines the
582
4.6 Packages and Programs
number of expansion terms used to
represent the solution.";
Begin@"`Private`"D;
Potential@boundary_, R_, alpha_, n_D :=
Block@8listed = 8<, int, boundaryh<,
Hreplace the independent variable
in the input by PhiLboundaryh =
boundary Й. f_@x2_. x1_D ▒ f@x2 phiD;
Hcalculate the coefficients
of the expansion d_nL
int = Integrate@boundaryh Sin@m Pi phi Й alphaD,
8phi, 0, alpha<D R ^ Hm Pi Й alphaL 2 Й alpha;
Do@AppendTo@listed, If@m m 0, 0, intDD, 8m, 0, n<D;
Hcalculate the
potential by using the sumL
pot = Sum@listed@@n1 + 1DD r ^ Hn1 Pi Й alphaL Sin@n1 Pi phi Й alphaD, 8n1, 0, n<D;
Htransform the potential to
cartesian coordinatesL
pot1 = pot Й. 8r ▒ Sqrt@x ^ 2 + y ^ 2D,
phi ▒ ArcTan@x, yD<;
Hgraphical representation of the
potential by ContourPlotL
ContourPlot@ pot1 Boole@x2 + y2 ├ R2 && y > 0 &&
y ├ Tan@alphaD xD, 8x, 0.0001, R<, 8y, 0, R< ,
PlotPoints > 200, ColorFunction > Hue,
Contours ▒ 15, PlotRange > All, Epilog >
8Line@880, 0<, 8R Cos@alphaD, R Sin@alphaD<<D<D
D;
End@D;
EndPackage@D;
4.6.3 Penning Trap
This package integrates the equations of motion for the Penning trap.
BeginPackage["Penning`"];
Clear[V,PenningI,PenningCMPlot];
4. Electrodynamics
583
PenningI::usage = "PenningI[r0_,z0_,e0_,n_,l_,te_]
determines the numerical
solution of the equation of motion for the relative
components. To integrate
the equations of motion, the initial conditions r0 =
r(t=0), z0 = z(t=0) and
the total energy e0 are needed as input parameters.
The momentum with respect
to the r direction is set to pr0=0. Parameters l and
n determine the
shape of the potential. The last argument te
specifies the end point of
the integration.";
PenningCMPlot::usage =
"PenningCMPlot[x0_,y0_,x0d_,y0d_,w_] gives a
graphical
representation of the center of mass motion for two
ions in the Penning trap.
The plot is created for a fixed cyclotron frequency
w in cartesian
coordinates (x,y,z). x0, y0, x0d, and y0d are the
initial conditions for
integration.";
Begin["`Private`"];
(* --- potential --- *)
V[x_, y_, l_, n_] := (x^2 + l^2*y^2)/2 + n^2/(2*x^2)
+
1/(x^2 + y^2)^(1/2);
(*--- numerical integration of the relative motion
---*)
PenningI[r0_,z0_,e0_,n_,l_,te_]:=Block[{intk,pz0},
(* --- initial value of the momentum in z direction
--- *)
pz0 = Sqrt[2*(e0-V[r0,z0,l,n])];
(* --- numerical solution of the initial value
problem --- *)
intk = NDSolve[{pr'[t] == n^2/r[t]^3 - r[t] +
r[t]/(r[t]^2+z[t]^2)^(3/2),
pz'[t] == -l^2*z[t] +
z[t]/(r[t]^2+z[t]^2)^(3/2),
r'[t] == pr[t],
z'[t] == pz[t],
584
4.6 Packages and Programs
(* --- initial values --- *)
r[0] == r0, z[0] == z0, pr[0] == 0, pz[0]
== pz0},
{r,z,pr,pz},{t,0,te}, MaxSteps->6000];
(* --- graphical representation --- *)
(* --- plot the potential --- *)
Show[
Block[{$DisplayFunction=Identity},
{Plot3D[V[x,y,l,n]-0.4,{x,-2,2},{y,-2,2},Mesh->False,
PlotPoints->25],
(* --- plot the tracks by ParametricPlot3D --- *)
ParametricPlot3D[Evaluate[{r[t],z[t],V[r[t],z[t],l,n]
} /. intk],
{t,0,te},PlotPoints->1000,
AxesLabel->{"r","z","V"}]}
],
AxesLabel->{"r","z","V"},
Prolog->Thickness[0.001],
ViewPoint->{1.3,-2.4,2}
]
];
(* --- center of mass motion in the Penning trap --*)
PenningCMPlot[x0_,y0_,x0d_,y0d_,w_]:= Block[{w0, a1,
b1},
(* --- fix parameters Omega_0 = 1.0 --- *)
w0 = 1.0;
a1 = 0.25;
b1 = 0.0;
If[w <= 2*w0,Print[" "];
Print[" cyclotron frequency too small"];
Print[" choose w > 2"],
(* --- determine the amplitudes from the initial
conditions --- *)
gl1 = 2*ar + 2*br - x0 == 0;
gl2 = -2*ai - 2*bi - y0 == 0;
gl3 = 2*bi*w1 + 2*ai*w2 - x0d == 0;
gl4 = 2*br*w1 + 2*ar*w2 - y0d == 0;
result =
Flatten[N[Solve[{gl1,gl2,gl3,gl4},{ar,ai,br,bi}]]];
(* --- solutions for the center of mass motion --- *)
4. Electrodynamics
585
x = 2*br*Cos[w1*t] + 2*bi*Sin[w1*t] + 2*ar*Cos[w2*t]
+ 2*ai*Sin[w2*t];
y = 2*ar*Sin[w2*t] - 2*ai*Cos[w2*t] + 2*br*Sin[w1*t]
+ 2*bi*Cos[w1*t];
z = a1*Cos[Sqrt[2 w0]*t + b1];
(* --- define frequencies --- *)
w1 = wc/2 + Sqrt[(wc/2)^2 - w0];
w2 = wc/2 - Sqrt[(wc/2)^2 - w0];
(* --- substitute the results result into the
variables x, y, and z --- *)
x = Simplify[x /. result];
y = Simplify[y /. result];
x1 = x /. wc -> w;
x2 = y /. wc -> w;
x3 = z /. wc -> w;
(* --- plot the solution --- *)
ParametricPlot3D[{x1,x2,x3},{t,0,60},AxesLabel->{"x",
"y","z"},
PlotPoints->1000,
Prolog->Thickness[0.001]]
]];
End[];
EndPackage[];
5
Quantum Mechanics
5.1 Introduction
Quantum mechanics compared with mechanics is a very young theory. The
theory emerged at 1900 when Max Planck (see Figure 5.1.1) examined the
blackbody radiation in thermodynamics. The discovery by Planck was that
the blackbody radiation can be described by a unified relation interpolating
between the high-frequency limit proposed by Wien and the low-frequency
limit favored by Rayleigh. The major assumption by Planck was that the
energy in this relation is linear in frequency and discrete HE = я wL. Planck
believed that this quantization applied only to the absorption and emission
of energy by matter, not to electromagnetic waves themselves. However, it
turned out to be much more general than he could have imagined.
588
5.1 Introduction
Figure 5.1.1.
Max Planck: born April 23, 1858; died October 4, 1947.
Another anchorman in quantum mechanics was Erwin SchrЖdinger (see
Figure 5.1.2) who invented wave mechanics in 1926. Reading the thesis of
Louis de Broglie, he was inspired to write down a wave equation which
established a second approach to mathematically describe quantum
mechanics.
Figure 5.1.2.
Erwin SchrЖdinger: born August 12, 1887; died January 4, 1961.
5. Quantum Mechanics
589
It was Werner Heisenberg (see Figure 5.1.3) who first gave a sound
description of quantum mechanics with his matrix mechanics in 1925.
Heisenberg was studying a set of quantized probability amplitudes when
he used a matrix algebra. These amplitudes formed a noncommutative
algebra. It was Max Born and Jordan in GЖttingen who recognized this
noncommutative algebra to be a matrix algebra. Another fundamental
achievement by Heisenberg in 1927 was the uncertainty principle which
governs all quantum mechanical systems.
Figure 5.1.3.
Werner Heisenberg: born December 5, 1901; died February 1, 1976.
Today, quantum mechanics is a central theory in physics to describe micro
and nano phenomena in atomic systems or semiconductors, for example.
Quantum mechanics in its field-theoretic extensions is important in
discussions of the unification of fundamental forces. The application of
quantum mechanics ranges from nano systems up to large-scale systems
such as black holes. Quantum mechanics is, in terms of its application, by
no means a self-contained theory. The major open question in quantum
theory is the unification with the theory of gravitation.
The current chapter introduces basic concepts of wave functions and
demonstrates the application of the SchrЖdinger equation to different
examples. In Section 5.2 the SchrЖdinger equation is introduced. Section
5.3 is concerned with the one-dimensional quantum dot model. Section 5.4
discusses the harmonic oscillator as a basic system to carry out quantum
590
5.1 Introduction
mechanical calculations. The harmonic oscillator is extended to an
anharmonic oscillator, which is important in the solution of nonlinear field
equations. Section 5.6 discusses the motion of a particle in a central force
field. The last section is concerned with the calculation of the second virial
coefficient and its quantum mechanical correction.
5.2 The SchrЖdinger Equation
The development of quantum mechanics as a field of study required an
equation that would adequately describe experimentally observed quantum
mechanical properties, such as the spectroscopic properties of atoms and
molecules. In 1926, SchrЖdinger wrote down the equation of motion for a
complex field in close analogy to the eikonal equation of optics [5.1].
Today, it is known as the SchrЖdinger equation. The SchrЖdinger equation
for a single particle reads
я2
еееее DyHxВ?, tL + V HxВ?L yHxВ?, tL,
i я yt = - ееее
2m
(5.2.1)
В?, tL denotes the wave function, V Hx
В?L is an external potential
where yHx
representing the source of forces in the quantum system, я is Planck's
constant, and m the mass of the particle under consideration.
The SchrЖdinger equation is a linear equation. It is well known that linear
partial differential equations allow a superposition of their solutions to
construct general solutions. Using this information with the two solutions
y1 and y2 of the SchrЖdinger equation (5.2.1) allows us to construct the
solution y = c1 y1 + c2 y2. We can identify SchrЖdinger's equation as a
diffusion equation if we define an imaginary diffusion constant. To solve
SchrЖdinger's equation, we can use, in principle, the same solution
procedure as for the diffusion equation. For certain initial values and
known boundary values, we find the evolution of the wave function y by
Eq. (5.2.1).
The main problem at the outset of quantum mechanics was the
interpretation of the wave function y. Although SchrЖdinger's linear
В?, tL
equation of motion (5.2.1) is completely deterministic, its solution yHx
is not a measurable quantity. In fact, the only observable quantities in
5. Quantum Mechanics
591
quantum mechanics are the probability y*y and any mean value based on
the distribution function y denoted by Xy ╩ Q ╩ y\.
Another consequence of the linearity of the SchrЖdinger equation is the
property of dispersion. It is well known that linear equations of motion
have dispersive waves as solutions. Since SchrЖdinger's equation (5.2.1)
contains an imaginary factor i, we can expect the solutions for a free
particle to undergo oscillations in the time domain. Plane waves are the
В?L = 0
simplest solutions to y. A particular solution of Eq. (5.2.1) with V Hx
is given by
В?, tL = ееееееее1еееееееее ei IkВ? Вx?-wHkL tM .
yk Hx
Х!!!!!!!! 3
I 2p M
(5.2.2)
The superposition of this particular solution delivers the general solution by
В?
В?, tL = ееееееее1еееееееее
i Ik Вx?-wHkL tM 3
d k.
yHx
Х!!!!!!!! 3 ?53 AHkL e
I 2pM
(5.2.3)
For simplicity's sake, we limit our consideration to one spatial dimension.
The solution (5.2.3) of the SchrЖdinger equation (5.2.1) is known as a
wave packet. The spectral density AHkL of the packet is completely
determined by the initial condition yHx, t = 0L = y0 HxL. The representation
(5.2.3) follows from the Fourier transform of the initial condition
╤
1
2p
AHkL = ееееееее
еееее ?-╤ y0 HxL e-i k x d x.
Х!!!!!!!!
(5.2.4)
Inserting the spectral density into the general solution (5.2.3), we get the
representation
╤
1
2p
╤
ееее!е ?-╤ ?-╤ y0 Hx'L ei HkHx-x'L-wHkL tL dk dx'
y Hx, tL = ееееееее
Х!!!!!!!
╤
= ?-╤ y0 Hx'L GHx, x', tL dx',
(5.2.5)
where the Green's function G is defined by
╤
1
ееее
eiH kH x-x'L-wHkL tL dk.
GHx, x', tL = ееее
2 p ?-╤
(5.2.6)
The dispersion relation wHkL of a dispersive wave is given by the defining
equation of motion. For the SchrЖdinger equation with vanishing external
potential V HxL = 0, the dispersion relation is wHkL = я k 2 Й H2 mL. Assuming
a localized distribution y0 HxL = dHxL for the initial condition of the wave
function, we can write the related solution as follows:
1
╤
ееее
ei kHx-a k tL dk.
yHx, tL = ееее
2 p ?-╤
(5.2.7)
592
5.2 S-Equation
This initial condition (assumed to derive the wave function y) cannot be
normalized. Although this assertion contradicts the quantum mechanical
interpretation, our only interest here is to show the dispersive behavior of
the wave function. The constant a = я Й H2 mL is purely numerical. The
relation (5.2.7) represents a solution of the SchrЖdinger equation (5.2.1)
for the case of a free particle located at x = 0 with t = 0. Since the
SchrЖdinger equation describes dispersive phenomena, we can observe a
broadening of the wave packet diminishing for t ь ╤. Its shape is
Х!!!!!!!
studied in the following. Replacing k by k = k К a t in Eq. (5.2.7), we
obtain
1
1
╤
yHx, tL = ееееееее
ееее ееееееее ? ei IkК
Х!!!!!!!
a t 2 p -╤
Х!!!!!!!!
a t H x-k2 LM
dk .
(5.2.8)
Computing the square in the exponent, we get
1
1
╤
ееее ееееееее ei x ЙH4 a tL ?-╤ e-i IxК
yHx, tL = ееееееее
Х!!!!!!!
at 2p
Х!!!!!!!
Substituting G = x К I2 a t M - k gives us
y Hx, tL =
2
╤
Х!!!!!!!!!!!! 2
4 a t -kM
dk .
(5.2.9)
ееееееееХ!!!!!!!
е1еееееееее ei x ЙH4 a tL ?-╤ ei G dG
2
at
2
1
= ееееееее
ееееееееее eiHx ЙH4 a tL+pЙ4L .
Х!!!!!!!!!!!
2 apt
2p
2
(5.2.10)
This representation of the wave function for a free particle can be used to
determine the probability of locating the particle at a certain time. As
discussed earlier, y is not a function directly observable by experiment. To
locate a particle at a certain location at a certain time, we have to study the
probability distribution ╩ y ╩2 of the particle. The probability distribution of
solution (5.2.10) is given by the expression
1
еееее .
╩ yHx, tL ╩2 = ееееееее
4apt
(5.2.11)
This result shows that the probability of finding a free particle as described
by SchrЖdinger's equation vanishes as time goes on. The probability of
finding a particle at any location decreases with time and vanishes as
t ь ╤. The dispersion process of the particle can be represented using
Mathematica in a sequence of pictures. To animate the dispersion process,
we first define the wave function y of the free particle:
5. Quantum Mechanics
593
Psi@x_, t_, hbar_: 1, mass_: 1D :=
Block@8alpha<, alpha = hbar Й H2 massL;
Exp@I Hx ^ 2 Й H4 alpha tL + Pi Й 4LD Й H2 Sqrt@alpha t PiDLD
where mass m and я are set to unity. By an appropriate scaling of the
coordinates, we can eliminate these constants in the equation of motion.
The probability distribution ╩ y ╩2 in relation (5.2.11) is only a function of
time and does not show any spatial dependence. However, if we examine
the wave function itself, we observe the spatial dispersion of the wave.
In Figure 5.2.4 a time sequence of the real part of the wave function is
plotted. The pictures are created by
Figure 5.2.4.
Re@PsiD
t=0.5
0.4
0.2
x
-6-0.2
-4-2 2 4 6
-0.4
Re@PsiD
t=1.
0.4
0.2
x
-0.2
-6
-4
-2 2 4 6
-0.4
Re@PsiD
t=1.5
0.3
0.2
0.1
x
-0.1
-6
- 4- 2 2 4 6
-0.2
-0.3
Re@PsiD
t=2.
0.2
0.1
x
-0.1
-6-0.2
-4-2 2 4 6
Re@PsiD
t=2.5
0.2
0.1
x
-0.1
-6
-4-2 2 4 6
-0.2
Re@PsiD
t=3.
0.2
0.1
x
-0.1
-6- 4- 2 2 4 6
-0.2
Time evolution of a wave packet for the SchrЖdinger equation. Initial conditions are
y0 HxL = dHxL.
594
5.2 S-Equation
The plots show that the amplitude of the wave function decreases from
about 0.5 to about 0.1 in a time range of 0.5 to 3.0. The dispersion of the
wave packet is observable in the wave function. The wave function
exhibits a reduced amplitude and a broadening of the initial packet.
The SchrЖdinger equation (5.2.1) not only describes time-dependent
properties of quantum mechanical systems but also stationary properties of
these systems. Contrary to our observations about free particles, we now
find that SchrЖdinger's equation describes stable particles. One central
question for such a system is how to uncover its intrinsic characteristics
such as the spectral properties. In the following, we examine one of the
fundamental models of quantum mechanics?the harmonic oscillator.
Before discussing the spectral properties of the harmonic oscillator, we
first summarize the solution steps for the time dependent SchrЖdinger
equation by a short graphical representation given in Figure 5.2.5.
1. Starting point of the solution procedure is the partial differential
equation (PDE) (5.2.1) and the initial solution of the wave function
yHx, 0L.
2. The use of the Fourier transform allows us to derive the spectral
density AHkL from the initial conditions.
3. A complete representation in Fourier space is attained when considering the time evolution, which is given by the dispersion relation wHkL.
4. The inversion of the representation in Fourier space delivers the
solution of the SchrЖdinger equation.
5. Quantum Mechanics
Figure 5.2.5.
595
Solution steps for a linear PDE by using the Fourier transform.
A similar solution procedure for nonlinear PDEs is discussed in Chapter 3
on nonlinear dynamics.
5.3 One-Dimensional Potential
In quantum mechanics, the measurement of a physical quantity A can
`
result only in one of the eigenvalues of the corresponding operator A. The
`
eigenvalues of A forming the spectrum of the operator might be discrete,
`
continuous, or both. The eigenfunctions of A form a complete basis that
can be used to expand an arbitrary wave function. The expansion
coefficients can be used to determine the probability of finding the system
`
in an eigenstate of the operator A with eigenvalue a. Central to quantum
mechanics is the determination of these eigenvalues and their related
eigenfunctions.
One of the fundamental quantities of a quantum dynamical system is its
energy. The operator corresponding to energy is the Hamiltonian operator
of the system. The Hamiltonian for a particle with mass m located in a
`
potential V is represented by H = -я2 Й H2 mL D + V HxL. The determination
of eigenvalues and eigenfunctions is demonstrated with a one-dimensional
model, the potential well. The potential well of depth V = -V0 discussed
in the following extends between -a ╖ x ╖ a where a is the maximum
596
5.3 One-Dimensional Potential
extension. Beyond the maximum extension, the potential vanishes. A
graphical representation of the potential is given in Figure 5.3.6.
Figure 5.3.6.
The potential well of depth V .
We study the case for which the kinetic energy of the particle is smaller
than the minimal potential value V0 (i.e., T < V0 ). The total energy E of
the system is E = T - V0 < 0. The particle has a negative total energy in
the domains 1 and 3 depicted in Figure 5.3.6. In classical mechanics, the
particle cannot be found in these regions. Contrary to classical mechanics,
however, quantum mechanics allows the existence of particles in regions
where they are classically forbidden. The domains 1 and 3 are governed by
`
the eigenvalue equations H y = E y, which are given in a differential
representation by
y'' - k2 y = 0,
(5.3.12)
where k2 = -2 m E Й я2 > 0 is a positive constant containing the total
energy. Primes denote differentiation with respect to the spatial coordinate.
The solution of Eq. (5.3.12) represents the domains 1 and 3 by
y1 = A1 ek x + B1 e-k x
y3 = A3 ek x + B3 e-k x
for
for
The related Mathematica result reads
-╤ < x ╖ -a,
a ╖ x < ╤.
(5.3.13)
(5.3.14)
5. Quantum Mechanics
597
s13 = DSolve@≥x,x \@xD N2 \@xD == 0, \, xD ЙЙ Flatten
8y ь Function@8x<, ?x k c1 + ?-x k c2 D<
In domain 2 the eigenvalue equation takes the form
y'' + k 2 y = 0,
(5.3.15)
where k 2 = 2 mHV0 + EL Й я2 > 0. The complete solution of (5.3.15) is given
by
y2 = A2 cos k x + B2 sin k x
for -a ╖ x ╖ a.
(5.3.16)
The computer algebra result is
s2 = DSolve@≥x,x \@xD + k2 \@xD == 0, \, xD ЙЙ Flatten
8y ь Function@8x<, c1 cosHk xL + c2 sinHk xLD<
From the normalization condition, it follows that the eigenfunctions given
by relations (5.3.13) and (5.3.14) require that the coefficients B1 and A3
vanish (i.e., B1 = A3 = 0). The remaining parameters A1 , B2 , A2 and B3 are
determined by applying the continuity condition of the wave function and
its first derivative at the end points of the potential well (x = -a and x = a).
The normalization condition requires
ps1 = \@xD Й. s13 Й. 8C@1D ▒ A1, C@2D ▒ B1< Й. B1 > 0
A1 ?x k
and
ps3 = \@xD Й. s13 Й. 8C@1D ▒ A3, C@2D ▒ B3< Й. A3 > 0
B3 ?-x k
598
5.3 One-Dimensional Potential
The conditions on the domain boundaries read
y1 = y2
y2 = y3
and
and
y'1 = y'2
y'2 = y'3
for
for
x = -a,
x= a
(5.3.17)
(5.3.18)
which can be given as
eq1 = ps1 == H\@xD Й. s2 Й. 8C@1D ▒ A2, C@2D ▒ B2<L Й. x ▒ a
A1 ?-a k Ц A2 cosHa kL - B2 sinHa kL
eq2 =
≥x ps1 == H≥x \@xD Й. s2 Й. 8C@1D ▒ A2, C@2D ▒ B2<L Й. x ▒ a
A1 ?-a k k Ц B2 k cosHa kL + A2 k sinHa kL
eq3 = ps3 == H\@xD Й. s2 Й. 8C@1D ▒ A2, C@2D ▒ B2<L Й. x ▒ a
B3 ?-a k Ц A2 cosHa kL + B2 sinHa kL
and
eq4 =
≥x ps3 == H≥x \@xD Й. s2 Й. 8C@1D ▒ A2, C@2D ▒ B2<L Й. x ▒ a
-B3 ?-a k k Ц B2 k cosHa kL - A2 k sinHa kL
The four equations form a homogeneous system of equations for the
unknowns A1 , B3 , A2 , and B2 . In a matrix representation, we get
-k a
sinHk aL 0
-cosHk aL
ij e
jj -k a
jj k e
-k
sinHk
aL
-k
cosHk aL 0
jjj
jj 0 -cosHk aL
-sinHk aL
e-k a
jj
j
-k cosHk aL -k e-k a
k 0 k sinHk aL
yz ij A1 yz
zz jj zz
zz jj A2 zz
zzz jjj zzz = 0.
zz jj B1 zz
zz jj zz
zj z
{ k B2 {
(5.3.19)
5. Quantum Mechanics
599
A nontrivial solution of Eq. (5.3.19) exists if the determinant of the matrix
vanishes. This condition delivers the relation
k2 - k 2 + 2 k k cotH2 k aL = 0
(5.3.20)
det1 =
Map@Coefficient@H8eq1, eq2, eq3, eq4< Й. Equal@a_,
b_D :> a bL, #D &,
8A1, A2, B2, B3<D ЙЙ Transpose ЙЙ Det ЙЙ Simplify
?-2 a k H2 k k cosH2 a kL + Hk2 - k 2 L sinH2 a kLL
with solutions
k = k tanHk aL,
k = -k cotHk aL.
(5.3.21)
(5.3.22)
spectral = MapAll@PowerExpand@#D &, Simplify@
Flatten@Solve@det1 == 0, NDDDD ЙЙ FullSimplify
8k ь -k cotHa kL, k ь k tanHa kL<
If we consider the first of these relations (5.3.21), we find that B2 = 0,
B3 = A1 , and A2 cos k a = A1 e-k a . The second relation, (5.3.22), results in
the conditions A2 = 0, B3 = -A1 , and B2 sin ka = -A1 e-k a .
sol1 = Solve@8eq1, eq2, eq3, eq4< Й. spectralP1T,
8A1, B2, A2, B3<D ЙЙ Simplify ЙЙ Flatten
Solve::svars : Equations may not give solutions for all "solve" variables. More?
8A1 ь -B3, B2 ь B3 ?a k cotHa kL cscHa kL, A2 ь 0<
600
5.3 One-Dimensional Potential
sol2 = Solve@8eq1, eq2, eq3, eq4< Й. spectralP2T,
8A1, A2, B2, B3<D ЙЙ Simplify ЙЙ Flatten
Solve::svars : Equations may not give solutions for all "solve" variables. More?
8A1 ь B3, A2 ь B3 ?-a k tanHa kL secHa kL, B2 ь 0<
We can thus distinguish between two systems of eigenfunctions: a
symmetric one and an antisymmetric one. The symmetry of the
eigenfunctions is obvious if we exchange the coordinates by x ь -x. The
symmetrical case is represented by
k = k tanHk aL,
y1 = A1 ek x ,
cosHk xL
y2 = A1 e-k a ееееееее
еееееееее ,
cosHk aL
-k x
y3 = A1 e
(5.3.23)
(5.3.24)
(5.3.25)
(5.3.26)
\1s = ps1 Й. sol2 Й. spectralP2T
B3 ?k x tanHa kL
\2s = \@xD Й. s2 Й. 8C@1D ▒ A2, C@2D ▒ B2< Й. sol2 Й.
spectralP2T
B3 ?-a k tanHa kL cosHk xL secHa kL
\3s = ps3 Й. sol2 Й. spectralP2T
B3 ?-k x tanHa kL
The antisymmetric case follows from the relations
k = -k cotHk aL,
y1 = -A1 ek x ,
sinHk xL
y2 = A1 e-k a ееееееее
ееееееее ,
sinHk aL
-k x
y3 = A1 e
(5.3.27)
(5.3.28)
(5.3.29)
(5.3.30)
5. Quantum Mechanics
601
\1a = ps1 Й. sol1 Й. spectralP1T
-B3 ?-k x cotHa kL
\2a = \@xD Й. s2 Й. 8C@1D ▒ A2, C@2D ▒ B2< Й. sol1 Й.
spectralP1T
B3 ?a k cotHa kL cscHa kL sinHk xL
\3a = ps3 Й. sol1 Й. spectralP1T
B3 ?k x cotHa kL
From the normalization condition
╤
-a
a
╤
2
2
2
2
?-╤ y dx = ?-╤ y1 dx + ?-a y2 dx + ?a y3 dx,
(5.3.31)
we get a relation for the undetermined amplitude A1
2
1
1
k
k
ееее
ее = a e-2 k a J1 + ееее
еее + ееее
еееее + ееее
ее N.
ka
k2 a
k2
A2
1
(5.3.32)
Relation (5.3.32) is satisfied for both the symmetric and antisymmetric
eigenfunctions.
To
calculate
the
eigenvalues,
note
that
k2 + k 2 = 2 m V0 Й я2 > 0 is independent of the total energy E. If we
introduce the parameter
V
0
2
2
2
C2 = a2 2 m ееее
я2ее = Hk + k L a ,
(5.3.33)
we can eliminate k from the eigenvalue equations. The equations
determining the eigenvalues are now
"#######################
C2 -Hk aL2
ееееееееееееееее
еееееееееее = tanHk aL,
ka
ka
- ееееееееееееееее
еееееееееее = tanHk aL.
"#######################
C2 -Hk aL2
(5.3.34)
(5.3.35)
Using relation (5.3.34) or (5.3.35), we can calculate k a and
E = я2 k 2 - 2 m V 0 .
602
5.3 One-Dimensional Potential
The problem with the potential well is not the derivation of its solution but
the calculation of the eigenvalues determined by Eqs. (5.3.34) and
(5.3.35). In the package QuantumWell`(see Section 5.8.1), we solve the
problem numerically for varying well depths V0 and well widths a.
Because the two determining equations of the eigenvalues are transcendent
equations, we have to switch to numeric calculations. The left-hand and
right-hand sides of Eqs. (5.3.34) and (5.3.35) are graphically represented
in Figure 5.3.7 for V0 = 12 and a = 1.
4
2
1
2
3
4
5
6
7
k
-2
-4
Figure 5.3.7.
Graphical representation of the eigenvalue equation for V0 = 12 and a = 1. The solid curves
represent the symmetrical case and the dashed curves represent the antisymmetric case. The
right-hand side of the eigenvalue equation reads tan k a.
Figure 5.3.7 is created by means of the function Spectrum[12,1] defined
in the package QuantumWell`. Also defined in the package
QuantumWell` are the eigenfunctions PsiSym[] and PsiASym[]. The
function Spectrum[] provides us with a graphical representation of the
eigenfunctions and prints out the related eigenvalues in a list. Some
examples of these eigenfunctions are given in Figures 5.3.8 and 5.3.9.
Function Spectrum[] creates a sequence of eigenfunction pictures starting
with the symmetric ones followed by the antisymmetric ones. Figures
5.3.8 and 5.3.9 contain the superposition of these sequences into one
picture.
ys
5. Quantum Mechanics
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
k1 =1.3018
k2 =3.8185
-2
Figure 5.3.8.
ya
603
-1
0
x
2
The symmetric eigenfunctions for a potential well with depth V0 = 12 and width a = 1. For
the given potential depth, there are a total of four eigenvalues, two of which are shown in
this figure and the other two are shown in the next figure. The solid eigenfunction with a
broad single maximum and no nodes is related to the lowest eigenvalue k =1.30183 of the
symmetric case. The second symmetric eigenvalue is k =3.81858. The corresponding
eigenfunction is dashed.
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
k1 =2.5856
k2 =4.8515
-2
Figure 5.3.9.
1
-1
0
x
1
2
The antisymmetric eigenfunction for the potential with V0 = 12 and a = 1. The two
antisymmetric eigenfunctions are correlated with the eigenvalues k =2.5856 and k =4.85759.
The first eigenfunction is represented by a solid curve and the second is dashed.
The sequence of eigenfunctions and eignvalues for different potential
depths V0 are generated with the function Spectrum[]. For a potential
depth of V0 = 44 with a potential with a = 2 we find
604
5.3 One-Dimensional Potential
Spectrum@44, 2D
15
10
5
2
-5
4
6
8
10
12
14
-10
-15
ys
ki = 0.745615
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-4
-2
0
2
4
k
5. Quantum Mechanics
605
ys
ki = 3.72294
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
x
-4
-2
0
2
4
ys
ki = 3.72294
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
x
-4
-2
0
2
4
ys
ki = 5.20377
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
x
-4
-2
0
2
4
606
5.3 One-Dimensional Potential
ys
ki = 6.67289
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
x
-4
-2
0
2
4
ys
ki = 8.11658
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
x
-4
-2
0
2
4
ya
ki = 1.49099
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
x
-4
-2
0
2
4
5. Quantum Mechanics
607
ya
ki = 2.97996
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
x
-4
-2
0
2
4
ya
ki = 4.46439
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
x
-4
-2
0
2
4
ya
ki = 5.94032
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
x
-4
-2
0
2
4
608
5.3 One-Dimensional Potential
ya
ki = 7.39956
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
x
-4
0
-2
2
4
ya
ki = 8.81407
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
x
-4
-2
0
2
4
eigenvalues sym eigenvalue k1 = 0.745615 asym eigenvalue k1 = 1.49099
sym eigenvalue k2 = 3.72294 asym eigenvalue k2 = 2.97996
sym eigenvalue k3 = 3.72294 asym eigenvalue k3 = 4.46439
sym eigenvalue k4 = 5.20377 asym eigenvalue k4 = 5.94032
sym eigenvalue k5 = 6.67289 asym eigenvalue k5 = 7.39956
sym eigenvalue k6 = 8.11658 asym eigenvalue k6 = 8.81407
5. Quantum Mechanics
609
The result is a system allowing 12 eigenvalues corresponding to 6
symmetric and 6 antisymmetric eigenfunctions.
5.4 The Harmonic Oscillator
The potential energy for a stable system exhibits a local minimum. One of
the standard methods of physics is to expand the potential energy around
the point of a local minimum in a Taylor series,
?
1 ≥2 V ???
x2 + ....,
J еееееееее N
V = V0 + ееее
2 ≥ x2 ???? x = 0
(5.4.36)
where x denotes the displacement from the equilibrium point. The
potential satisfies ≥V Й ≥ x = 0 at the stable equilibrium point. If the
particle of mass m only undergoes small oscillations around the
equilibrium point, the first two terms of relation (5.4.36) are sufficient to
describe the potential energy. Choosing the origin of the energy to be
identical with V H0L of the expansion, we can express the Hamiltonian of
the harmonic oscillator
2
p
(5.4.37)
ееее + ееееk2 x2 ,
Hcl = ееее
2 еm
??
?
is the spring constant of the oscillator. We
where k = ≥2 V Й ≥ x2 ???
?? x = 0
already know that the classical solution for the harmonic oscillator is given
by a periodic function
%
xHtL = A cosHw t + bL where w = $%%%%%%%
еееее
m
k
(5.4.38)
and the system undergoes harmonic oscillations around the equilibrium
point. The time average of the total energy follows from relations (5.4.37)
and (5.4.38)
610
5.4 Harmonic Oscillator
XE\T =
Z
ccccccccc
2S
2S
2
i ccccccc
Z ii m H≥t x@tDL
k
y
j
j
j
j
j
z
j
j
cccccccccccccccc
cccccccc
c
ccccc
c + cccc x@tD2 z
j
z Й. x > Function@t,
j
?
j
j 0
2
2
kk
{
k
y
y
z
2
z
z
A Cos@Z t + EDDz
z е t ЙЙ Simplifyz
z
z Й. k > Z m
{
{
1
еееее A2 m w2
2
XE\T = ееее
m A2 w2 = m w2 ЙЙx 2 ,
2
1
(5.4.39)
where T denotes the period of the oscillation; that is, the time-averaged
energy depends quadratically on the amplitude A of the oscillations.
In this section, our aim is to examine the quantum mechanical properties of
the harmonic oscillator and compare them with the classical situation. The
transition from classical to quantum mechanics is formally achieved by
replacing the classical coordinates with quantum mechanical operators:
x ь x` and p ь p` = я Й i ≥x . Using the transformations in the Hamiltonian
yields the timeless SchrЖdinger equation in the form of an eigenvalue
problem given by
d2
еееее J ееее
d x2
2
2
w m
2mE
ееееееее
еееее x2 + ееееееее
еееее N yHxL = 0,
я2
я2
(5.4.40)
where y denotes the set of eigenfunctions of the Hamiltonian. By an
Х!!!!!!!!!!!!!!!
appropriate scaling of the spatial coordinate x = m w Й я x and of the
eigenvalue ╤ = 2 E Й Hя wL, we get the eigenvalue problem in a standard
form
d2
еееее - x2 + ╤N yHxL = 0.
J ееее
d x2
(5.4.41)
eigenValueEquation = ≥[,[ \@[D [2 \@[D + ≥ \@[D == 0
-yHxL x2 + ╤ yHxL + yёё HxL Ц 0
The question here is what type of function yHxL satisfies Eq. (5.4.41). As a
solution, we try the expression
yHxL = vHxL e-x Й2 .
2
(5.4.42)
5. Quantum Mechanics
611
[2
c
2 E
ansatz = \ > FunctionA[, v@[D ф cccccccc
x2
y ь FunctionBx, vHxL ?- ееее2ее е F
From Eq. (5.4.41), it follows that the amplitude v has to satisfy the ODE
v'' - 2 x v' + H╤ - 1L vHxL = 0,
(5.4.43)
transformedEVeq =
eigenValueEquation Й. ansatz ЙЙ Simplify
x2
?- ееее2ее е HH╤ - 1L vHxL - 2 x vё HxL + vёёHxLL Ц 0
where primes denote differentiation with respect to x. To be physically
acceptable, the wave function yHxL must be continuous and finite. The
amplitude vHxL defined by Eq. (5.4.43) is a finite function if v is a
polynomial of finite order.
solution = DSolve@transformedEVeq, v, [D ЙЙ Flatten
1-╤ 1 2
:v ь FunctionB8x<, c1 H ееее╤-1
ееееее е ; еееее ; x NF>
е2ееееее HxL + c2 1 F1 J ееееееее
4
2
This type of solutions exists if
╤ = 2 n + 1, where n = 0, 1, 2, ....
(5.4.44)
For each value n there exists a polynomial of order n which satisfies Eq.
(5.4.43). These polynomials are known as Hermite polynomials, defined by
2
dn
2
-x
Hn HxL = H-1Ln ex ееее
d еxеееnе e .
(5.4.45)
In Mathematica, the Hermite polynomials are identified by the function
HermiteH[]. The solutions of the eigenvalue problem become with the
eigenvalues
612
5.4 Harmonic Oscillator
eigenValues = ≥ ▒ 2 n + 1
╤ ь 2n+1
a two-component solution determined by c1 and c2 , the integration
constants
ve = v@[D Й. solution Й. eigenValues
n 1
c1 Hn HxL + c2 1 F1 J- ееееее ; еееее ; x2 N
2 2
it is known that the hypergeometric function 1 F1 is divergent for x ь ■╤.
Thus, we can chose c2 = 0. The eigenfunctions thus are determined by
ve = v > Function@[, $vD Й. $v > Hve Й. C@2D > 0L
v ь Function@x, c1 Hn HxLD
The eigenfunctions thus can be written
ps = \@xD Й. ansatz Й. ve
x2
?- ееее2ее е c1 Hn HxL
where c1 is a constant determined by the normalization. The wave function
y of the harmonic oscillator is represented in scaled coordinates by
1
yn HxL = ееееееееееееееее
еееееееее Hn HxL e-x Й2 .
"#####################
n Х!!!! #
n! 2
2
p
(5.4.46)
The corresponding eigenvalues of the harmonic oscillator are
1
En = я w In + ееее
2 M.
(5.4.47)
Each eigenvalue has its own eigenfunction which is either even or odd
with respect to coordinate reflections in x. Note that the eigenvalues and
5. Quantum Mechanics
613
eigenfunctions have a one-to-one correspondence (i.e. the spectrum is
non-degenerate). The first four even and odd eigenfunctions of the
harmonic oscillator are depicted in figures 5.4.10 and 5.4.11.
The probability distribution ╩ y ╩2 of finding the harmonic oscillator in a
certain state n in the range x ■ d x is given by
╩ y ╩2 d x =
еееееееееn1еееееее
ееее Hn2 HxL e-x d x = wqm HxL d x.
Х!!!!
2
n! 2
p
(5.4.48)
The classical probability of finding a particle in the range x ■ d x is
determined by the period T of the oscillator.
dt
x
w
ееее d ееее
ееее ,
wclHxL = ееееTее = ееее
2p
╩v╩
(5.4.49)
where xHtL is represented by the classical solution (5.4.38). The
corresponding velocity v follows from the time derivative of x:
x 2
е L %.
1 - H ееее
v = -A w $%%%%%%%%%%%%%%%%%%%%
A
(5.4.50)
In scaled variables x we find for the classical probability the relation
wclHxL =
1
1
ееееееееееееееее
еееееееее ееееееееееееееее
еееееееееееееееее! .
Х!!!!!!!!!!!!!!
Х!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
2
2p
2 n+1
1-x ЙH2 n+1L
(5.4.51)
Specifying either the energy or the eigenvalue of the harmonic oscillator
enables us to compare the classical probability with the quantum
mechanical result. A graphical representation of these two quantities is
given in Figures 5.4.12 and 5.4.13. Figure 5.4.12 shows the ground state
and Figure 5.4.13 shows the eigenvalue with n = 5. It can be clearly seen
that the quantum mechanical behavior of the probability density is
different from its classical behavior. In the classical case, the particle
spends most of its time near the two turning points, where the density ╩ y ╩2
is large. Quantum mechanically, there is a high probability that the particle
is located near the center of the potential (ground state). In an excited
state, we observe regions where the particle cannot be found (see Figure
5.4.13). This is due to the fact that the quantum mechanical probability
density oscillates for n > 0, which, in turn, is a consequence of the
oscillations of the wave function.
At the classical turning points, a completely different behavior of the
quantum particle is apparent. Where the classical particle cannot be found
614
5.4 Harmonic Oscillator
in quantum mechanics, there is a finite probability for locating a particle
outside the potential well. This tunneling of the particle into the potential
barrier is unusual and cannot be explained by classical mechanics.
The eigenfunctions and the harmonic potential V HxL are superimposed on
each other in Figures 5.4.10 and 5.4.11. The related classical and quantum
mechanical probabilities are shown in Figures 5.4.12 and 5.4.13. The
functions to create these figures for certain eigenvalues are contained in
the package HarmonicOscillator` (see Section 5.8.2).
V, y
10
8
6
4
2
-4
Figure 5.4.10.
-2
2
4
x
Symmetric eigenfunctions of the harmonic oscillator VHxL = x2 for eigenvalues
n = 0, 2, 4, 8. The eigenfunctions are centered around the energetic levels
E = я w Hn + 1 Й 2L corresponding to the eigenvalues n.
5. Quantum Mechanics
615
V, y
10
8
6
4
2
-4
Figure 5.4.11.
2
-2
4
x
Antisymmetric eigenfunctions of the harmonic oscillator V HxL = x2 for eigenvalues
n = 1, 3, 5, 9. The eigenfunctions are centered around the energy levels E = я w Hn + 1 Й 2L
corresponding to the eigenvalues n.
wkl , wqm
1.2
1
0.8
0.6
0.4
0.2
-2
Figure 5.4.12.
-1
1
2
x
Classical and quantum mechanical probability density for the harmonic oscillator in the
ground state. The classical probability shows a singular behavior at the turning points of
the motion.
616
5.4 Harmonic Oscillator
wkl , wqm
0.5
0.4
0.3
0.2
0.1
-4
Figure 5.4.13.
-2
2
4
x
Comparison between the classical and quantum mechanical probability density for the
eigenvalue n = 5. The singular points of the classical probability wcl are located at
x = ■ 3.316.
The given derivation of the wave function is based on the defining
equation of the Hermite polynomials (5.4.41). The solution of the scaled
equation (5.4.41) delivers the complete set of eigenfunctions in one step.
In the following, we show how the set of eigenfunctions can be derived by
an iterative procedure involving creation and annihilation operators a+ and
a- . All of the eigenfunctions are created out of the ground state of the
harmonic oscillator,
1
p
-x Й2
.
y0 HxL = ееее
4 еееее e
Х!!!!
2
(5.4.52)
The whole set of eigenfunctions can be created using the following
creation and annihilation operators a+ and a- , which act in the spatial and
momentum space:
`
1
≥
1
a+ = ееее
еееее Ix - ееее
еее M = ееее
еееее Ix - i p` M,
Х!!!!
Х!!!!
(5.4.53)
≥x
2
2
`
1
≥
1
`
еееее Ix + ееее
еее M = ееее
еееее Ix + i pM.
a- = ееее
Х!!!!
Х!!!!
(5.4.54)
≥x
2
2
The name of the operators stems from the action of the wave functions
respectively creating and annihilating a quantum mechanical state. The
actions of operators a+ and a- can be demonstrated by introducing two
functions aminus[] and across[]. The definitions are given below and use
the representations of Eqs. (5.4.53) and (5.4.54).
5. Quantum Mechanics
617
1
aminus@\_, [_: [D := cccccccccc H[ \ + ≥[ \L
Х!!!!
2
1
across@\_, [_: [D := cccccccccc H[ \ ≥[ \L
Х!!!!
2
If we apply the defined functions to the ground state, we get the first
excited state or, simply, zero. The definition of the ground state is
contained in the function yn .
1
[2
\n_ @[_D := cccccccccccccccc
cccccccccccccccc HermiteH@n, [D E cccc2cc
"########################
Х!!!!#
n ! 2n S
We get from the application of the generating operator
across@\0 @[DD
Х!!!! - ееееx22ее е
x
2 ?
ееееееееееееееее
ееееееееееееееееее
4
Х!!!!
p
The anhilation operator appled to the ground state gives
aminus@\0 @[DD
0
Comparing the Mathematica result with the first excited state y1 , we find
that they are equivalent.
across@\0 @[DD == \1 @[D ЙЙ Simplify
True
618
5.4 Harmonic Oscillator
Х!!!!!!
This is also true if we incorporate the factor n ! on the right-hand side
for higher n. The higher eigenfunctions are derived from the ground state
by the relation
1
n!
+ n
yn HxL = ееееееее
Х!!!!!!еее Ha L y0 HxL.
(5.4.55)
Repeatedly applying an operator is achieved by using the function Nest[].
Nest@across, \0 @[D, 5D ЙЙ Simplify
Х!!!! - ееееx22ее е
x H4 x4 - 20 x2 + 15L
2 ?
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееее
4
Х!!!!
p
We assume that yn is a function of x. When using Nest[], we can
repeatedly apply the function across[] to the wave function Psi[]. The
number of applications of across[] to yn is controlled by the second
argument of Nest[]. In the above example, we applied across[] five times
to yn . The result is the representation of y5 . If we are interested in the
functions preceding y5 , we can use NestList[] instead.
\List = NestList@across, \0 @[D, 5D ЙЙ Simplify
x2
x2
Х!!!! - ееееx22ее е
Х!!!! - ееееx22ее е
x ?- ееее2ее е H2 x2 - 1L
x H2 x2 - 3L
?- ееее2ее е
2 ?
2 ?
ееееееееееееееее
ееееееее
ееее
е
еее
е
е
ееееееееееееееееееееееееееееееее
ееееееее
ееее
е
е
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееее ,
: ееееееее
ееее
е
еее
е
,
,
,
4
4
4
4
Х!!!!
Х!!!!
Х!!!!
Х!!!!
p
p
p
p
x2
Х!!!! - ееееx22ее е
?- ееее2ее е H4 x4 - 12 x2 + 3L
x H4 x4 - 20 x2 + 15L
2 ?
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееее
ееееееее
ееее
е
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееее >
,
4
4
Х!!!!
Х!!!!
p
p
The unnormalized wave functions contained in the list yList are
eigenfunctions of the harmonic oscillator. To determine the normalization
factors, we integrate yList over the total space:
5. Quantum Mechanics
norm = 1 М
619
┬
"####################################################################################
#
Map@H?┬ # е [L &, Expand@\List2 DD
1
1
1
1
91, 1, ееееееее
еееее , ееееееее
еееее , ееееееее
ееееееееее , ееееееееееееееее
ееее!ее =
Х!!!!
Х!!!!
Х!!!!
Х!!!!!!
2
6 2 6 2 30
The normalized eigenfunctions are now given by
\List = \List norm
x2
x2
x2
Х!!!! - ееееx22ее е
x ?- ееее2ее е H2 x2 - 1L ?- ееее2ее е x H2 x2 - 3L
?- ееее2ее е
2 ?
ееееееееееееееееее , ееееееееееееееееееееееееееееееее
ееееееееееееее , ееееееееееееееееееееееееееееееее
еееееееееееееееееее ,
: ееееееее
еееееееее , ееееееееееееееее
4
4
4
4
Х!!!!
Х!!!! Х!!!!
Х!!!! Х!!!!
Х!!!!
p
2 p
3 p
p
x2
x2
?- ееее2ее е H4 x4 - 12 x2 + 3L ?- ееее2ее е x H4 x4 - 20 x2 + 15L
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееееееее , ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееее >
4
4
Х!!!!
Х!!!!
Х!!!!!!
! Х!!!!
2 6 p
2 15 p
The preceding functions are collected in the package HarmonicOscillator`. A complete listing is contained in Section 5.8.2.
5.5 Anharmonic Oscillator
So far, we have discussed problems which assume harmonic particle
motion. In real systems, harmonic motion is the exception rather than the
rule. In general, forces are not proportional to linear displacements. From
the example of the pendulum in classical mechanics (see Section 2.4.8.6),
we recall that the restoring force is not proportional to linear
displacements. Another example is that of large molecules in quantum
chemistry: In contrast to the binding potential of a diatomic molecule
[5.2], the forces between atoms in a large molecule are anharmonic.
The classical work on anharmonic forces in quantum mechanics was
initiated by PЖschel and Teller [5.3], who examined the single anharmonic
oscillator. Lotmar [5.4] in 1935 studied an ensemble of anharmonic
oscillators and established their connection with large molecules. We
examine here an altered PЖschel?Teller potential, which today is used in
the inverse scattering method of solving nonlinear evolution equations (see
620
5.5 Anharmonic Oscillator
Chapter 3). The interaction potential for a quantum mechanical system was
given by FlЭgge [5.5] in the form
V HxL = -V0 sech2 x,
(5.5.56)
where V0 is a constant determining the depth of the potential well. The
related stationary SchrЖdinger equation in scaled variables reads
d2
еееее + l + V0 sech2 xN yHxL = 0.
J ееее
d x2
(5.5.57)
PTEVproblem = ≥x,x \@xD + HO + V0 Sech@xD2 L \@xD == 0
HV0 sech2 HxL + lL yHxL + yёё HxL Ц 0
In our examination, we determine the eigenvalues l = 2 m E Й я2 , which
depend on the potential depth V0 . Another point of our study is the form of
the wave functions in the asymptotic range ╩ x ╩ ь ╤. We first introduce
some changes in the notation of Eq. (5.5.57). Substituting for the
independent variable x using the relation x = tanhHxL in Eq. (5.5.57), we
can carry out the transformation by
t1 = PTEVproblem Й. \ ▒ Function@y, \@[@yDDD
yёёHxHxLL xё HxL2 + HV0 sech2 HxL + lL yHxHxLL + yё HxHxLL xёёHxL Ц 0
then we replace the new dependent variable x by
t2 = t1 Й. [ > Function@x, Tanh@xDD
yёёHtanhHxLL sech4 HxL 2 tanhHxL yё HtanhHxLL sech2 HxL + HV0 sech2 HxL + lL yHtanhHxLL Ц 0
Using the inverse of the hyperbolic tan, we get
5. Quantum Mechanics
621
t3 = t2 Й. x > ArcTanh@[D Й. O > O
2
yёёHxL H1 - x2 L - 2 x yё HxL H1 - x2 L + HH1 - x2 L V0 - lL yHxL Ц 0
which in traditional representation is
dy
d
H1 - x2 L ееее
ееее IH1 - x2 L ееееd еxеее M + Hl + V0 H1 - x2 LL y = 0 where
dx
-1 < x < 1,
(5.5.58)
or the equivalent standard representation of Eq. (5.5.58)
dy
d
l
ееее
ееее IH1 - x2 L ееее
ееее M + IV0 + ееееееее
ееее M y = 0.
dx
dx
1-x2
(5.5.59)
Equation (5.5.59) is the defining equation for the associated Legendre
polynomials, which is checked by the line
solution = DSolve@t3, \, [D ЙЙ Flatten
Х!!!!!
Х!!!!!
:y ь FunctionB8x<, c1 P ее1ее lIХ!!!!!!!!!!!!!!!!!
HxL + c2 Q ее1ее lIХ!!!!!!!!!!!!!!!!!
HxLF>
!
!
4 V +1 -1M
4 V +1 -1M
2
0
2
0
A graphical check of the two Legendre polynomials shows that Legendre
Qnm is divergent at the boundaries,
622
5.5 Anharmonic Oscillator
Plot@Evaluate@
H\@[D Й. solution Й. 8V0 ▒ N HN + 1L, O > n2 <L Й.
8N > 25, n > 2, C@1D > 0, C@2D > 1<D,
8[, 1, 1<, AxesLabel > 8"[", "\"<D;
y
600
400
200
-1
0.5
-0.5
1
x
-200
-400
whereas the Legendre Pnm is finite at the boundaries,
Plot@Evaluate@
H\@[D Й. solution Й. 8V0 ▒ N HN + 1L, O > n2 <L Й.
8N > 4, n > 2, C@1D > 1, C@2D > 0<D,
8[, 1, 1<, AxesLabel > 8"[", "\"<D;
y
10
7.5
5
2.5
-1
-0.5
-2.5
-5
-7.5
0.5
1
x
5. Quantum Mechanics
623
Thus, for a finite solution of the PЖschel?Teller problem we have to
assume that c2 = 0. The solution then becomes
solutionPT = solution Й. C@2D > 0
Х!!!!!
Х!!!!!
:y ь FunctionB8x<, c1 P ее1ее lIХ!!!!!!!!!!!!!!!!!
HxL + 0 Q ее1ее lIХ!!!!!!!!!!!!!!!!!
HxLF>
!
!
4 V +1 -1M
4 V +1 -1M
0
2
2
0
For the solution of Eq. (5.5.59), we assume, in addition, that the potential
depth is given by positive integer V0 = N HN + 1L, where N is a positive
number. Equation (5.5.59) possesses discrete bound solutions in the range
x ? @-1, 1D if and only if l = -n2 < 0 with n = 1, 2, ..., N. The
eigenfunctions of the SchrЖdinger equation (5.5.59) are proportional to the
associated Legendre functions PnN HxL defined mathematically by
nЙ2
PnN HxL = H-1Ln H1 - x2 L
n
d
ееее
еееее P HxL,
d xn N
(5.5.60)
where PN HxL are the Legendre polynomials of degree N:
1
dN
N
PN HxL = ееееееее
еееее ееееееееее Hx2 - 1L .
N! 2N d x N
(5.5.61)
The constant connecting the Legendre functions with the eigenfunctions of
the PЖschel?Teller problem is a product of the normalization condition
and the eigenfunctions. The following function represents the
eigenfunctions of the PЖschel?Teller system. The associated Legendre
polynomials are given by the function LegendreP[].
624
5.5 Anharmonic Oscillator
PoeschelTeller[x_, n_Integer, N_Integer] :=
Block[{norm, integrand, xi},
If[n <= N && n > 0,
(* --- the associated Legendre polynomial specify
the eigenfunction --- *)
integrand = LegendreP[N, n, xi];
(* --- determine the normalization constant --- *)
norm = Integrate[integrand^2/(1-xi^2), {xi, -1,
1}];
(* --- normalize the eigenfunctions --- *)
integrand = integrand/Sqrt[norm] /. xi ->
Tanh[x];
Simplify[integrand],
(* --- check errors in the input parameters --- *)
If[N<n,
Print["--- wrong argument n > N"]];
If[n<0,
Print["--- wrong argument n < 0"]]]
]
The eigenfunctions for N = 4 are
Table[PoeschelTeller[x,i,4],{i,1,4}]
1 Х!!!! Х!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
9 еееее 5 coshH2 xL + sinhH2 xL HtanhHxL - 1L tanhHxL H7 tanh2 HxL - 3L,
4
1 Х!!!!!!!!!! Х!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
1 Х!!!!
ееееее 5 H3 coshH2 xL - 4L sech4 HxL, - ееееее 105 coshH2 xL + sinhH2 xL
4
4
1
35
HtanhHxL - 1L2 tanhHxL HtanhHxL + 1L, ееееее $%%%%%%%%%
еееееееее % sech4 HxL=
4
2
The results for n = 1 and n = 3 are graphically represented in Figure 5.5.14:
5. Quantum Mechanics
625
y
0.75
0.5
0.25
-4
-2
-0.25
2
4
x
-0.5
-0.75
Figure 5.5.14.
Eigenfunctions of the modified PЖschel?Teller potential for discrete eigenvalues n = 1
(solid) and n = 3 (dashed) at N = 4.
So far we derived the discrete spectrum of the modified PЖschel?Teller
problem. In the following we consider the continuous eigenvalues
l = k 2 > 0 of the stationary SchrЖdinger Eq. (5.5.59). The eigenfunctions
thus read
1-x 2
ееее N
yHx; kL = aHkL J ееееееее
4
-i kЙ2
2 F1
Х
1+x
IaХ , b; cХ ; ееее2еееее M,
(5.5.62)
Х
Х!!!!!!!!!!!!!!!!!!!!
Х!!!!!!!!!!!!!!!!!!!!
and
where aХ = 1 Й 2 - i k + V0 + 1 Й 4 , b = 1 Й 2 - i k - V0 + 1 Й 4
cХ = 1 - i k are constants depending on the model parameters and the
eigenvalues. The label 2 F1 denotes the Gaussian hypergeometric function.
Х!!!!!!!!!!!!!!
In the limit x ь ╤ sechHxL = 1 - x2 = 2 ex Й H1 + e2 x L ~ 2 e-x and the
solution reduces to the form y ~ aHkL e-i k x . The explicit representation in
the limit x ь -1 of the solution (5.5.62) is given by
Х
aХ b
ееее H1 + xL + OHx2 LN.
yHx; kL = aHkL e-i k x J1 + ееее
2 cХ
(5.5.63)
The asymptotic expansion of the hypergeometric function 2 F1 is carried
out by first replacing the argument ееее12 H1 + xL with z and then by expanding
2 F1 up to first order around z = 0
626
5.5 Anharmonic Oscillator
Series[Hypergeometric2F1[a,b,c,z],{z,0,1}]
abz
1 + ееееееееееееее + OHz2 L
c
Hence, the leading term in the asymptotic representation of the
eigenfunction y for x ь -╤ is
y ~ aHkL e-i k x .
(5.5.64)
In the other limit x ь ╤, we first transform the hypergeometric function
using the linear transformation 2 F1 Ha, b, c, zL = d 2 F1 Ha, b, c, 1 - zL,
yielding
2 F1
ij
1
1
jj ееее
,
V0 + ееее
jj 2 - i k + $%%%%%%%%%%%%%%%%%
4
k
yz
1+x
1-x
1
ееее12 - i k - $%%%%%%%%%%%%%%%%%
V0 + ееее
; 1 - i k; ееее2еееее zzz = I ееее2еееее M
4
z
ik
{
2 F1
y
ij
1+x zz
1
1
1
1
jj ееее
$%%%%%%%%%%%%%%%%%
z
V0 + ееее
,
ееее
+
V
+
ееее
;
1
i
k;
ееее
е
еее
е
jj 2 - $%%%%%%%%%%%%%%%%%
0
4
2
4
2 zz
{
k
1-x i k
= I ееее2еееее M
ij
y
i
1-x z
1
1
1
1
jj F jjj ееее
V0 + ееее
, ееее
+ $%%%%%%%%%%%%%%%%%
V0 + ееее
; 1 + i k; ееее2еееее zzzz Ъ
jj 2 1 jj 2 - $%%%%%%%%%%%%%%%%%
4
2
4
k
{
k
(5.5.65)
GH1+i kL GH-i kL
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееее1ееееее +
"###############
"###############
1
1
1
GJ ееее2 -i k+ V0 + ееее4 N GJ ееее2 -i k- V0 + ееее4 N
1-x i k
I ееее2еееее M
2 F1
1 - i k;
.
ij
1
1
1
1
jj ееее
V0 + ееее
, ееее
+ $%%%%%%%%%%%%%%%%%
V0 + ееее
;
jj 2 - $%%%%%%%%%%%%%%%%%
4
2
4
k
zy
1+x z
GH1-i kL GHi kL
ееее
еееее z Ъ ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ее
2 zz
1 "###############
1
1 "###############
1
y
{ GJ ееее2 + V0 + ееее4 N GJ ееее2 - V0 + ееее4 N {
If the potential depth is of the form V0 = N HN + 1L, we observe that
Х!!!!!!!!!!!!!!!!!!!!
1 Й 2 - V0 + 1 Й 4 is always a negative integer. Since the function G is
5. Quantum Mechanics
627
singular for these points, the second term on the right hand side always
vanishes. Taking this into account (5.5.65) reduces to
2 F1
ij
1
1
jj ееее
V0 + ееее
jj 2 - i k + $%%%%%%%%%%%%%%%%%
4 ,
k
.
yz
1+x
1-x
1
ееее12 - i k - $%%%%%%%%%%%%%%%%%
V0 + ееее
; 1 - i k; ееее2еееее zzz = I ееее2еееее M
4
z
ik
{
(5.5.66)
y
ij
1-x zz
1
1
1
1
j ееее
$%%%%%%%%%%%%%%%%%
zЪ
V0 + ееее
,
ееее
+
V
+
ееее
;
1
+
i
k;
ееее
е
еее
е
jj 2 - $%%%%%%%%%%%%%%%%%
2 F1 j
0
4
2
4
2 zz
{
k
GH1+i kL GH-i kL
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееее1ееееее
"###############
"###############
1
1
1
GJ ееее2 -i k+ V0 + ееее4 N GJ ееее2 -i k- V0 + ееее4 N
In the limit x ь ╤, the wave function y has the representation
y ~ e-i k x + bHkL ei k x ,
(5.5.67)
where bHkL is the reflection coefficient of the wave. Relation (5.5.67)
means that an incoming wave of amplitude 1 is reflected by a part
determined by bHkL.
An asymptotic expansion of the hypergeometric function for x ь 1
consequently gives us the representation in the form
y ~ aHkL
GH1+i kL GH-i kL
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееее1ееееее e-i k x .
"###############
"###############
1
1
1
GJ ееее2 -i k+ V0 + ееее4 N GJ ееее2 -i k- V0 + ееее4 N
(5.5.68)
Comparing relation (5.5.68) with (5.5.67), we observe that the reflection
coefficient of the wave vanishes. The transmission coefficient aHkL in the
case V0 = N HN + 1L takes the form
aHkL =
GJ ееее12 -i k+"###############
V0 + ееее14 N GJ ееее12 -i k-"###############
V0 + ееее14 N
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееее .
GH1+i kL GH-i kL
(5.5.69)
A wave is free of reflection if the potential takes the form V = V0 sechHxL
and the depth of the potential is an integer number V0 = N HN + 1L.
For V0 = N HN + 1L, the entire calculation procedure can be activated by
AsymptoticPT[] which is part of the package AnharmonicOscillator`(see Section 5.8.3). By calling AsymptoticPT[] we get the
asymptotic representation of the eigenfunction in the limits x ь ■╤. The
628
5.5 Anharmonic Oscillator
results of the expansion are contained in the global variables w1a and
w2a. Function AsymptoticPT[] can also handle the case in which N is an
integer. In addition to the eigenfunction, function AsymptoticPT[]
delivers information about the reflection and transmission coefficients ╩ b ╩2
and ╩ a ╩2 . These two characteristic properties of the scattering problem
satisfy ╩ a ╩2 + ╩ b ╩2 = 1. PlotPT[], which is also part of the package
AnharmonicOscillator`, gives a graphical representation of the reflection
and transmission coefficients. This function plots five curves for different
k values. The range of the k values is specified as first and second
arguments in the function PlotPT[]. The third argument of PlotPT[]
determines the coefficient. We can choose between two types of
coefficient. Whereas "t" will create a plot for the transition coefficient, the
"r" string will create the reflection plot. Two examples for k ini = 0.05 and
kend = 0.5 are given in Figures 5.5.15 and 5.5.16. The pictures are created
by
PlotPT@0.05, .5, "r"D;
╩b╩2
1
0.8
0.6
0.4
0.2
1.2
Figure 5.5.15.
1.4
1.6
1.8
2
N
The reflection coefficient ╩ b ╩2 is plotted as a function of N. The ensemble of curves
represent the reflection coefficient for energy values k in the interval k ? @0.05, 0.5D for
N ? @1, 2D. The top curve represents the value k =0.05. The other k values > 0.05 follow
below the top curve.
and
PlotPT@0.05, .5, "t"D;
5. Quantum Mechanics
629
╩a╩2
1
0.8
0.6
0.4
0.2
1.2
Figure 5.5.16.
1.4
1.6
1.8
2
N
The transmission coefficient ╩ a ╩2 of the PЖschel?Teller potential is plotted across the
depth parameter N of the potential. The energy values k are taken from the interval
k ? @0.05, 0.5D for N ? @1, 2D. The lowest curve corresponds to k =0.05.
The structure represented in Figures 5.5.15 and 5.5.16 is repeated in each
of the intervals 8N, N + 1 ╩ N ╔ 1<. Two neighboring intervals for a
potential depth ranging between V0 = 2 and V0 = 6 (N = 1 and N = 2) are
represented in Figure 5.5.17. In this figure, the reflection coefficient is
shown for a range of k values by means of a surface plot. The pictures are
created by the sequence
th = AsymptoticPT@NN, kkD;
Plot3D@Evaluate@thP2TD, 8NN, 1, 3<,
8kk, 0.05, 0.75<, AxesLabel ▒ 8"N", "k", "╩b╩2
"<,
PlotPoints > 30, Mesh > FalseD;
Plot3D@Evaluate@thP1TD, 8NN, 1, 3<, 8kk, 0.05, 0.75<,
AxesLabel ▒ 8"N", "k", "╩a╩2
"<,
PlotPoints > 30, Mesh > FalseD;
630
5.5 Anharmonic Oscillator
1
0.75
╩b╩2 0.5
0.25
0
1
0.6
0.4k
1.5
N
0.2
2
2.5
3
1
2 0.75
╩a╩
0.5
0.25
0
1
0.6
0.4k
1.5
N
0.2
2
2.5
3
Figure 5.5.17.
The reflection and transmission coefficient is plotted as a function of N and k. The values
for the potential depth are taken from N ? @1, 3D and the energy interval is k ? @0.05, 0.75D.
We observe that the reflection coefficient decreases as the energy increases. On the other
hand, the transmission coefficient increases with the increase in energy.
A collection of functions examining the anharmonic PЖschel-Teller
potential is contained in the package AnharmonicOscillator`. Useful
5. Quantum Mechanics
631
functions in examining the anharmonic model are PoeschelTeller[],
AsymptoticPT[] and PlotPT[] (compare the complete listing in Section
5.8.3).
5.6 Motion in the Central Force Field
The stationary states of a particle in a spherically symmetric potential are
determined by the SchrЖdinger equation with the Hamiltonian operator
`
я2
(5.6.70)
ееее ?2 +V HrL,
H = - ееее
2 еm
Х!!!!!!!!!!!!!!!!!!!!!!!!!!
where r = x2 + y2 + z2 measures the distance of the particle from the
origin of the potential. Using the spherical symmetry of the problem, we
can rewrite the SchrЖdinger equation in spherical coordinates
я2
1
≥2
еееее ееее еееееееее r A- ееее
2 m r ≥ r2
я
1
≥
1
ееееееее
еееее J ееееееее
ее ееее≥еее sin J ееее
еее + ееееееее
еееее ееее≥еееее N +
2 m r2 sin J ≥ J
≥J
sin2 J ≥ f2
2
2
(5.6.71)
V HrL - EE yHr, J, fL = 0,
or, in a more compact form,
`2
я2 1 ≥2
я2
J- ееее
еееее ееее еееееееее r + ееееееее
еееее L + V HrL - EN y = 0,
(5.6.72)
2 m r ≥ r2
2 m r2
`2
where L is the square of the angular momentum operator. Problems which
can be identified by such a Hamiltonian operator are very common in
physics such as follows:
1. The H-atom
2. An ion with one electron
3. The three-dimensional harmonic oscillator
4. The three-dimensional potential well, quantum dot
5. The Yukawa particle (a shielded Coulomb potential)
6. The free particle.
In close analogy to classical motion in a central field, we find in quantum
mechanics that the angular momentum is conserved. The angular
momentum is defined by
632
5.6 Central Force Field
В? В? В?
L = x ╣ p.
(5.6.73)
Other constants of motion are the Hamiltonian, the square of the angular
momentum, and the z-component of the angular momentum. The related
`
` `2
operators H , L , and Lz create a complete system of commuting operators.
The solutions of the related eigenvalue problems completely determine the
properties of the system. As in classical mechanics, we can take advantage
of the conservation of angular momentum to reduce a three-dimensional
problem to a one-dimensional one. Similarly, we can use the conservation
of the angular momentum to separate the coordinates r, J, and f in the
SchrЖdinger equation (5.6.72).
The dependence of the wave function y on the angles J and f is
`
`2
determined by the operators L and Lz . In spherical coordinates, we can
`
express the z component of the angular momentum by Lz = -i я ≥f . The
`
eigenvalues of Lz , are found by solving the equation
≥yHfL
ееееяi ееееееее
еееее = Lz yHfL,
≥f
(5.6.74)
where 0 ╖ f ╖ 2 p. The solutions of Eq. (5.6.74) are
i
yHfL = A e еееея Lz f .
(5.6.75)
Since the solution (5.6.75) must be uniquely defined, it has to satisfy the
condition
yHfL = yHf + 2 pL.
(5.6.76)
The eigenvalues Lz Й я = m where m = 0, ■1, ■2, ... satisfy condition
`
(5.6.76). The eigenvalues of the operator Lz are thus discrete and
represented by
Lz = я m, where m = 0, ■1, ■2, ....
Since we require normalized eigenfunctions (i.e.,
normalized solutions are
1
2p
ym HfL = ееееееее
еееее ei m f
Х!!!!!!!!
(5.6.77)
2p
?0
y*m ym df = 1), the
(5.6.78)
A similar treatment yields the eigenvalues and eigenfunctions of the square
`2
of the angular momentum L from the differential equation
`2
L y = L2 y.
(5.6.79)
5. Quantum Mechanics
633
`2
In spherical coordinates, the operator L is represented by
`2
1
≥
L = -я2 J ееееееее
ее ееее≥еее sin J ееее
еее +
sin J ≥ J
≥J
2
1
ееееееее
еееее ееее≥еееее N.
sin2 J ≥ f2
(5.6.80)
Inserting expression (5.6.80) into Eq. (5.6.79), we get
1
≥
≥
≥2
1
L2
J ееееееее
ее еееееее sin J ееее
еее + ееееееее
еееее еееееееее + ееее
ее N yHJ, fL = 0.
sin J ≥ J
≥J
я2
sin2 J ≥ f2
(5.6.81)
Equation (5.6.81) is the defining equation of the spherical harmonics Yl,m
if the eigenvalues satisfy L2 = я2 l Hl + 1L with l = 0, 1, 2, ....:
1
≥
≥
≥2
1
J ееееееее
ее еееееее sin J ееее
еее + ееееееее
еееее еееееееее + l Hl + 1LN Yl,m HJ, fL = 0. (5.6.82)
sin J ≥ J
≥J
sin2 J ≥ f2
`2
The eigenvalues of L are determined by the quantum numbers
l = 0, 1, 2, .... Their related eigenfunctions are the spherical harmonics
Yl,m of order l. Comparing the structure of the eigenfunctions of the
harmonic oscillator to that of the eigenfunctions of the angular momentum
`2
`2
L , we observe that in the case of L with eigenvalues L2 = я2 l Hl + 1L,
there are 2 l + 1 eigenfunctions Yl,m . The eigen- functions Yl,m , however,
are different in the second quantum number m, which is known as the
magnetic quantum number. For a fixed value of L2 , m counts the different
projections on the z-axis. If we determine l, we find different values for m
m = 0, ■1, ■2, ..., ■l
(5.6.83)
and limited to the range -l ╖ m ╖ l. For the proof of the above relations,
we refer the reader to the book by Cohen-Tannoudji et al. [5.6].
The complete representation of the spherical harmonics for positive m is
H-1Lm
2p
H2 l+1L Hl-mL!
еееее ei m f $%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
ееееееееееееееее
еееееееееееее sinm J Pml Hcos JL.
Yl,m HJ, fL = ееееееее
Х!!!!!!!!
2 Hl+mL!
(5.6.84)
Pm
l HxL denotes the mth associated Legendre function of order l. In case of
negative quantum numbers m, we use the relation
*
HJ, fL.
Yl,-m HJ, fL = H-1Lm Yl,m
(5.6.85)
If we use the representation of the spherical harmonics given by relation
(5.6.84), it is easy to show that the Yl,m are also eigenfunctions of the
`
operator Lz . By a simple calculation, we find
≥
ееееяi ееее
ееее Y HJ, fL = я m Yl,m HJ, fL.
≥ f l,m
(5.6.86)
634
5.6 Central Force Field
We now can state that the spherical harmonics are eigenfunctions of both
the z-component of the angular momentum operator and the square of the
angular momentum operator. The corresponding eigenvalues are
L2 = я2 l Hl + 1L
and
Lz = я m.
(5.6.87)
The spherical harmonics are accessed in Mathematica by the function
SphericalHarmonicY[] available in the package CentralField` in Section
5.8.4. The Legendre polynomials are available using LegendreP[].
So far, we have determined the eigenfunctions depending on J and f.
Separating the angular terms from the radial part of the wave function, we
get the representation
yHr, J, fL = hHrL Yl,m HJ, fL.
(5.6.88)
Relation (5.6.88) used with Eq. (5.6.72) allows the derivation of a
determining equation for the radial part hHrL of the wave function y. The
wave function separates because the coordinate system of our problem is
separable. The radial function hHrL is dependent on the energy E, the
quantum number l, and the potential energy V HrL. Consequently, the radial
part of the wave function is independent of m: In a spherical potential,
there are no distinguishing directions breaking the symmetry.
Inserting relation (5.6.88) into the SchrЖdinger equation (5.6.72) and using
our above results for the angular momentum, we get, after substituting
uHrL = r hHrL, the eigenvalue problem for the radial part of the wave function
я2
d2
еееее еееееееее + V HrL +
J- ееее
2 m d r2
я l Hl+1L
ееееееее
ееееееееееее N uHrL = E uHrL.
2 m r2
2
(5.6.89)
uHrL = r hHrL is substituted since for r ь 0, the function hHrL has to be finite
(i.e., uHrL ь 0 for r ь 0). Note that in Eq. (5.6.89), all parameters are
known except for potential V HrL. For the following discussion, we assume
that the potential V HrL represents a Coulomb interaction of the two
particles,
Z e2
V HrL = - ееееrеееее .
(5.6.90)
This type of potential applies to the hydrogen and hydrogenlike atoms
where Z = 1 as well as to ionized atoms like He+ , Li2+ , and so forth.
5. Quantum Mechanics
635
The stationary states of an electron in a Coulomb potential result from the
eigenvalue equation
d2
еееее +
J ееее
d r2
l Hl+1L
2mE
2mZe
ееееееее
еееее + ееееееее
еееееееее - ееееееее
ееееее N uHrL = 0.
я2
я2 r
r2
2
(5.6.91)
To carry out our calculation, it is convenient to introduce scaled variables
r
r = ееее
a
E
╤ = ееее
ее ,
E0
and
(5.6.92)
is
Bohr's
radius
and
where
a = я2 Й Hm e2 L ╨ 5.29 ╣ 10-11 m
E0 = e2 Й H2 aL = m e4 Й я2 ╨ 13.5 eV, the ionization energy of the hydrogen
atom. The SchrЖdinger equation (5.6.91) is thus represented by
d2
2Z
ееееее + ╤ + ееееrееее J ееее
d r2
l Hl+1L
ееееееее
ееееее N uHrL = 0,
r2
(5.6.93)
which allows a representation as
2Z
l Hl + 1L y
i + cccccccc
radialEVProblem = ≥U,U u@UD + j
c cccccccccccccccc
cccccc z
z u@UD == 0
jH
U
U2
k
{
l Hl + 1L
2Z
J- ееееееееееееееее
ееееее е + e + еееееееееее N uHrL + uёёHrL Ц 0
r2
r
We restrict our calculations to the case of bound states characterized by
negative energy values. To find appropriate representations of a solution
ansatz for uHrL, we examine the limits r ь 0 and r ь ╤. The function uHrL
is either given by a polynomial in r ul HrL = ra H1 + a1 r + a2 r2 + ...L or by
an exponential relation ul = A e-g r +B eg r , where g2 = -╤. The results of
these expressions are conditions for the parameters a and B which satisfy
a = l + 1, B = 0. Using these results both expressions are reducible to
ul HrL = rl+1 e-g r f HrL
or, in a manageable form,
tr1 = u > Function@U, Ul+1 фJ U f@UDD
u ь Function@r, rl+1 ?-g r f HrLD
(5.6.94)
636
5.6 Central Force Field
Substituting expressions (5.6.94) into Eq. (5.6.93) and using x = 2 g r, we
get the standard form of Kummer's differential equation:
Z
е M f = 0,
x f '' + H2 Hl + 1L - xL f ' - Il + 1 - ееее
g
(5.6.95)
where primes denote differentiation with respect to x. The Mathematica
version of this transformation using original variables is gained by
g1 = radialEVProblem Й. tr1 ЙЙ Simplify
?-g r rl
HHr g2 - 2 Hl + 1L g + 2 Z + e rL f HrL + 2 Hl - g r + 1L f ё HrL + r f ёё HrLL Ц 0
The solution can be directly derived from
solution = DSolve@g1, f, UD ЙЙ Flatten
Х!!!
i - Х!!!
e l-бZ - e
Х!!! y
c1 U jjjj- ееееееееееееееееееееееееееееееее
еееееееее , 2 l + 2, 2 б e rzzzz +
Х!!!ееееееееееееееее
e
k
{
Х!!!!
Х!!!
Ig-б e M r
2 l+1
c2 L -Х!!!!e ! l-б Z-Х!!!!e! I2 б e rMF>
?
: f ь FunctionB8r<, ?Ig-б
Х!!!!
e Mr
ееееееееееееееее
ееееееееееееееее
Х!!!!! еееееееееее
e
which simplifies if we assume that the energies e are negative:
f@UD Й. solution Й. H > H ЙЙ PowerExpand ЙЙ Simplify
Х!!!!
e Mr
?Ig+
ij
Z
i
Х!!! y
Х!!! yz
2 l+1
jjc1 U jjjl + ееееееее
I-2 e rMzzz
Х!!!ееее + 1, 2 l + 2, -2 e rzz + c2 L-l- ееееХ!!!!
еZееее!ее -1
e
e
k
k
{
{
The solutions of Eq. (5.6.95) are, in general, confluent hypergeometric
functions (1 F1 )
Z
е , 2 l + 2; 2 g rM
fl HrL = c 1 F1 Il + 1 - ееее
g
(5.6.96)
reducing to Laguerre's and Kummer's function. To satisfy the
normalization condition, series (5.6.96) must terminate at a finite order.
5. Quantum Mechanics
637
This restriction excludes Kummer's function HC1 = 0L and induces the
quantization of the energy values by
Z
е = -nr ,
l + 1 - ееее
g
with
nr = 0, 1, 2, ....
(5.6.97)
The solution of Eq. (5.6.97) with respect to g delivers
Z
ееееееее ,
g = ееееееее
nr +l+1
(5.6.98)
or, by replacing g2 = -╤, yields energy values ╤ = -Z 2 Й Hnr + l + 1L2 to be
Z2
Hnr +l+1L
Z2
E = - ееееееееееееееееееее2е E0 = - ееее
ее E .
n2 0
(5.6.99)
The quantum number n is the principal quantum number determined by the
radial quantum number nr Hnr = 0, 1, 2, ...L and the angular quantum
number l Hl = 0, 1, 2, ?L. The wave function of the electron in the
Coulomb potential is given by
yn,l,m Hr, J, fL =
2Z
Nn,l rZ rЙn 1 F1 Il + 1 - n, 2 l + 2; ееееnееее rM Yl,m HJ, fL,
(5.6.100)
where Nn,l is the normalization constant
Hn+lL!
1
2Z
еееееееее $%%%%%%%%%%%%%%%%%%%%%%%%%%
ееееееееееееееее
еееееееее I ееее
ееее M
Nn,l = ееееееее
H2 l+1L!
2 n Hn-l-1L!
n
l+3Й2
(5.6.101)
.
The radial part of the wave function hHrL consists of
2Z
hn,l HrL = Nn,l rl e-Z rЙn 1 F1 Il + 1 - n, 2 l + 2; ееееnееее rM.
(5.6.102)
Since the first argument in the hypergeometric function is a negative
integer, the function 1 F1 in the radial part reduces to a polynomial known
as a Laguerre polynomial. In Mathematica, the Laguerre polynomials are
denoted by LaguerreL[]. One useful parameter of the radial wave
function is nr = n - l - 1. This parameter counts the nodes of the
eigenfunction along the horizontal axis. This behavior is shown in Figure
5.6.18 for n = 3 and l = 0, 1, 2. Figure 5.6.18 is created by
Plot@8Radial@r, 3, 0, 1D, Radial@r, 3, 1, 1D,
Radial@r, 3, 2, 1D<, 8r, 0, 25<,
AxesLabel ▒ 8"r", "h"<, Prolog ▒ Thickness@0.001DD;
638
5.6 Central Force Field
h
0.15
0.1
0.05
5
10
15
20
25
r
-0.05
Figure 5.6.18.
Radial part h of the wave function for n = 3 and l = 0, 1, 2.
The function Radial[] used in the Plot[] function is part of the package
CentralField`. This package also contains Angle[] for the angular part of
the wave function. The definition of Angle[] is, in some ways, redundant
since Mathematica accounts for the angular part of the wave function
under the name SphericalHarmonicY[]. However, we separately define
the angular part of the wave function to show how relations (5.6.84) and
(5.6.85) are expressed in terms of Mathematica.
The above wave function is applied to representations of orbitals of an
atom or a molecule. Chemists, for example, work with molecular orbital
theory to describe the binding of atoms. This theory makes extensive use
of the angular wave functions Yl,m . In order to describe the binding of a
molecule, it is necessary to use a linear combination of the angular parts of
the wave function. We create such a superposition of the Yl,m 's by the
function Orbital[], which is part of the package CentralField`. Orbital[]
creates sums and differences of the spherical harmonics in the form
wHJ, fL = H ╩ Y Ll,m ■ Yl,-m ╩2 .
(5.6.103)
Relation (5.6.103) represents the probability of finding an electron within
a certain domain of the angular part of the space. In Figures 5.6.19-22, we
have plotted some particular examples for orbitals.
5. Quantum Mechanics
-0.1
00.05
0.1 -0.05
0.1
0.05
.05
0
-0.05
5
-0.1
0.4
0.2
0
-0.2
-0.4
Figure 5.6.19.
Angular part of the wave function for l = 2 and m =0.
639
640
5.6 Central Force Field
0.05 -0.1
0.025
0
-0.025
0.025
-0.05
0.05
0
0.1
0.1
0
-0.1
-0
Orbital for the quantum numbers l =2 and m =╠1 formed from the difference ╩ Y2,1 - Y2,-1 ╩2 .
Figure 5.6.20.
0.05
0
-0.05
0.2
0
-0.2
0
-0.2
0.2
0
2
Figure 5.6.21.
A plot of the sum of the wave functions with quantum numbers
l =2 and m = ■2.
5. Quantum Mechanics
641
-0.1
-0.05
00.05
0.1
0.5
0.25
0
-0.25
-0.5
Figure 5.6.22.
0.1
0.05
0
-0.05
-0.1
Representation of the orbital ╩ Y ╩2 for quantum numbers l =3 and m =0.
Figures 5.6.19-22 show an inner view of the orbitals for a certain range of
f. Similar pictures for other quantum numbers are created by the
superposition of the angular wave functions Yl,m with the help of Angle[].
The figures of the orbitals are created by the function sequence
642
5.6 Central Force Field
AnglePlot[Orbital[l,m,J,-f,''plus''],J,f] . An example of the appli- cation
of this function is given below.
AnglePlot@Orbital@T, I, 4, 2, "minus"D, T, ID;
0.1-0.1
0
0
0.1
-0.1
0.2
0.1
0
-0.1
-0.2
5.7 Second Virial Coefficient and Its Quantum
Corrections
Nearly 100 years ago, Kannerligh Onnes discribed the thermodynamic
behavior of a gas in form of an equation which should become as virial
equation of state one of the most successful theories for the link between
the microscopic physics of molecular interactions and macroscopic
thermodynamic properties:
BHTL
CHTL
DHT L
PV
ееее
еееее = 1 + ееееееее
ее + ееееееее
еее + ееееееее
ееее + ?,
RT
V
V2
V3
(5.7.104)
5. Quantum Mechanics
643
where BHTL, CHTL, and DHTL are the second, third, and fourth viral
coefficients of increasing complexity, R is the gas constant, V is the
volume, and T is the absolute temprature in the virial equation.
Immediately after the introduction of the virial equation, Ornstein
calculated the second virial coefficient (SVC) using Gibb's statistical
calculation techniques
╤
BHTL = -2 p NA ?
0
?-U HrLЙr
J ееееееее
ееееееее - 1N r2 ? r
k T
B
(5.7.105)
where NA is Avogadro's constant, U HrL is the intermolecular potential, and
kB is the Boltzmann constant. The exciting history of the virial equation
and its relation to the phenomenological van der Waals equation as well as
the history of the calculation of BHTL for various molecular potentials is
covered in an excellent article by Rowlinson [5.7]. He discusses the van
der Waals equation and its implications to the development of the real gas
and the liquid. In spite of the strong influence of the van der Waals
equation on the study of molecular interactions, it could not describe
accurately the behavior of any substance. Rowlinson points out how an
empirical proposal of Onnes was combined with the theoretical
development of Gibbs and Ornstein to produce the viral equation of state,
one of the most useful theories of any state of matter.
Before the theory was worked out completely and before the quantum
theory of the intermolecular potential was developed, the second virial
coefficient (SVC) was investigated by interaction potentials of the kind
A
B
ее - ееее
еM
U HrL = I ееее
rm
rn
(5.7.106)
mostly associated with Lennard-Jones [5.8]. After the derivation of the
dispersion forces proportional to r-6 by London [5.9], the H12 - 6Lpotential has become very popular. Theory and numerical results of this
and related potential are discussed in detail in the classical monographs by
Hirschfelder et al. [5.10] on the molecular theory of gases and liquids and
by Mason and Spurling [5.11] on the viral equation of state. As will be
pointed out subsequently, the SVC is an integral over a function of U HrL.
In teaching statistical thermodynamics, however, one wants to give a final
result not as an integral but as an explicit function of the temperature and
644
5.7 Second Virial Coefficient
molecular parameters. Especially for the Hm - nL-Lennard-Jones potential
(abbreviated by (m,n)-LJ) analytical results in terms of series expansions
with the G function have been given in [5.12]. It was pointed out, however,
by several authors, also in recent textbooks that for the potential,
especially the 12 - 6, no closed solution exists. That this statement is not
correct will be shown subsequently in the sketch on analytical approaches
to the SVC. What is lacking, however, is a consistent derivation of the
SVC, its quantum corrections, and its temperature derivatives from one
integral. The present section aims at such a unified derivation. Also, other
results in the literature will be reduced to these results.
5.7.1 The SVC and Its Relation to Thermodynamic Properties
The necessary formulas for the SVC and its quantum corrections are
collected to show the importance for thermodynamic functions. The virial
equation of state was given in Eq. (5.7.104). A knowledge of the virial
coefficients and their temperature dependence describes the pVT behavior
of the gas completely, if one assumes the convergence of the series. For
the classical part Bc of the BHTL, one derives
╤
Bc HTL = 2 p NA ?0 H?- b UHrL - 1L r2 ? r
2pN b
╤
dU
A
еее ?0 ?- b UHrL I ееееdrееее M r3 ? r
= - еееееееее3еееееее
(5.7.107)
after partial integration. NA is Avogadro's number, b = HkB TL-1 , kB is the
Boltzmann constant, and UHrL is the interatomic or intermolecular
potential. The index c on B denotes the purely classical part of our
considerations. For low temperatures and light atoms and molecules like
He, Ne, and H2 , one has to take quantum mechanics into account. It was
shown with the H12 - 6L potential for He that at very low temperatures, the
full quantum mechanical calculation has to be performed, but for
temperatures above 5K, the semiclassical expansion without the symmetry
term is sufficient:
я2
я2
2
ее B + J ееееmее N Bq2 + ?
B = Bc + ееее
m q1
(5.7.108)
with
3
Bq1 =
2
p NA b
-b U
ееееееее
HU 'L r2 ? r
6еееееееее ?0 ?
╤
(5.7.109)
5. Quantum Mechanics
645
and
4
Bq2 = -
p NA b
ееееееее
еееееееее
6
╤
?
0
HU''L2
HU'L2
?-b U A ееееееее
еееее + ееееееее
ееее +
10
5 r2
4
bHU'L
b HU'L
ееееееее
ееееее - ееееееее
еееееееее E r2 ? r.
9r
72
3
2
(5.7.110)
The SVC is important for the correct calculation of thermodynamic
functions at high temperatures, as it includes not only the bound states
usually only taken into account in the calculation of partition functions but
also meta-stable and continuum states. This was shown explicitly for a
Rydberg diatomic potential by Sinanoglu and Pitzer [5.13]; a more recent
discussion on the splitting of the phase space of the SVC was given by
Friend [5.14].
The thermodynamic functions related to the SVC, B, and its temperature
derivatives Bn = T n Hd n B Й dTn L are given by the internal energy
~
~0
B1
U -U
ееееееее
ееееее = -J ееее
~ее + ?N,
RT
V
(5.7.111)
the enthalpy
~
~0
B-B1
H -H
ееееееее
ееееее = ееееееее
~ еееее + ?,
RT
(5.7.112)
V
the entropy
~
~0
2
B1
S -S
B
ееееееее
ееее = -9ln p + ееее
~ее + ееееееее
~2ее ?=,
R
V
(5.7.113)
2V
and the specific heat
~
~0
C p -C p
HB-B1 L2
B2
ееееееее
ееееееее = -9 ееее
~ее - еееееееее~еееееееее + ?=
RT
V
V
(5.7.114)
mJT C0p = -@B - T B 'D +
2
1
RT
2
еееее
ее HB - T B'L B '' + ?E + ?.
~ A2 B - 2 T B B' - ееееееее
C0
V
(5.7.115)
p
Thermodynamic functions give the extent of the values from the value of a
perfect gas in its normal state denoted by a superscript (╟); the tilde (~)
represents molar quantities. From these formulas follows that for a
complete analytical theory of the SVC and for thermodynamic functions
with two-body interactions, one has to calculate
646
5.7 Second Virial Coefficient
BHTL = Bc HTL + Bq1 HTL + Bq2 HTL + ?
(5.7.116)
5.7.2 Calculation of the Classical SVC Bc HTL for the
H2 n - nL-Potential
A useful method of evaluating the thermodynamic properties of gases at
high temperatures is to treat the entire gas as a monoatomic assembly with
gas imperfections given by
BHTL
CHTL
DHT L
PV
ееее
RеTееее = 1 + ееееееее
V ее + ееееееее
V 2еее + ееееееее
V 3ееее + ?,
(5.7.117)
where BHTL, CHTL, and DHTL denote the second, third, and fourth virial
coefficients, respectively. Our interest here is the second virial coefficient
BHTL (SVC) and its quantum mechanical corrections up to second order.
All of the thermodynamic properties of the gas are then obtained directly
from the equation of state as represented by Eq. (5.7.117).
In the following calculations, we will examine the two-parameter
Lennard-Jones potential (LJ):
m
ддддд
n e дддддддд
s n
s m
n-m
LJ = 2 J ддддддддддддддддддддд N
JJ ддддддд N - J ддддддд N N
n-m
r
r
m
cccc
ccccccc
nH
V m
V n
m+n
2 I cccccccccccccccc M
II ccccc M + I ccccc M M
m + n
r
r
where e is the well depth and s is the internuclear distance. Our interest is
mainly concerned with the case when m = n and n is replaced by an even
number of m. As a two-parameter potential with e, s, the Hm - nL-potential,
is simple but not very flexible. An additional parameter is introduced in
the spherical Kihara hard-core potential [5.15]:
n
m
дддддддддддд д
s-2 a n
s-2 a m
2 n I дддд
дд M n-m e II дддддддд
дддддддд M - I дддддддд
дддддддд M M
r-2 a
r-2 a
m
Kihara = дддддддддддддддддддддддддддддддд
дддддддддддддддддддддддддддддддд
дддддддддддддддддддддддддддддддд
ддддддддддддддд
n-m
m
m
n
n cccccccccc
2 a+V
2 a+V
2 n H ccc
c L m+nc H IH cccccccc
ccccc L + H cccccccc
ccccc L M
m
2 a+r
2 a+r
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccccccc
m + n
5. Quantum Mechanics
647
The Kihara potential is ╤ for r < 2 a and is connected with the LJ
potential if we replace the radial coordinate r, the potential depth e, and
intermolecular distance by
me
m
ддддддддддддд
Hn - mL H дддддддд
ддддд L n-m
n-m
transforms = 9r ф 2 a + r, e ф дддддддддддддддддддддддддддддддд
дддддддддддддддд
ддддддддддд , s ф 2 a + s=;
n
Applying these transformations to the Kihara potential, we find
tK = Simplify@Kihara Й. transformsD
m
m
еееее
еееее
n ееееееее
m e ееееееее
s n
s m
n-m
n-m
2 I ееееее M
J ееееееееееееееееее N
JI ееееее M - I ееееее M N
m
r
r
n-m
Comparing the LJ potential with the transformed Kihara potential, we
observe their equivalence:
PowerExpand@tK == LJD
True
meaning that both potentials are identical. Thus, we can unify the
calculations for one type of potential. We therefore restrict our
considerations to the LJ potential. We note that the following results are
also valid in case of the Kihara potential. Our main interest is concerned
with a subclass of LJ potentials where the exponents Hn, mL are given by an
even integer and the integer itself. For such a combination, the LJ potential
reduces to a H2 n - nL-potential, which is given by
UHrL = LJ Й. 8n ф 2 n, m ф n<
s 2n
s n
4 e JI ееееее M - I ееееее M N
r
r
648
5.7 Second Virial Coefficient
The first derivative, the intermolecular force, needed to evaluate (5.7.107)
follows from the potential by differentiating UHrL with respect to r:
≥UHrL
Force = SimplifyA- ддддддддддддддддддддд E
≥r
s n
s n
4 n e H еееее
L H2 H еееее
L - 1L
r
r
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееееееееее
r
Inserting the potential U HrL and the force into Eq. (5.7.107), we find
?
1
Bc = дддддд H2 p N A bL ? ?-b UHrL Force r3 ? r
3
0
Integrate::gener : Unable to check convergence. More?
╤
s 2n
s n
2
s n
s n
i
y
еееее p b j? 4 ?-4 b e IH ееrеее L -H ееrеее L M n r2 e I ееееее M J2 I ееееее M - 1N ? rz N A
r
r
3
k 0
{
At first glance, the result is disappointing because Mathematica does not
evaluate the integral. However, it returns the integral containing the
explicit expressions for the potential U and its first derivative. A second
examination of the integral reveals that we found a Laplace transform of
the first derivative of U, the negative force. To recognize that the above
integral represents a Laplace transform, let us introduce the following
substitutions:
≥Hs t-1Йn L
ддддддддддддддддд DifferentialDHtL=;
substitution = 9r ф s t-1Йn , DifferentialDHrL ф дддддддддддддддд
≥t
Applying this substitution to the integrand Bc , we are able to reduce
(5.7.107) to a Laplace integral. The integrand of this integral is calculated
by the transformation
5. Quantum Mechanics
649
integrand = PowerExpandA
1
дддддд HH-2 p N A bLL ?-b UHrL Force r3 DifferentialDHrL Й. substitution Й.
3
DifferentialDHtL ф 1E
8
2
еееее ?-4 Ht -tL b e p t-3Йn H2 t - 1L b e s3 N A
3
Inserting the new integrand into the classical part of the SVC, we find
Bc = ?
?
integrand ? t
0
Integrate::gener : Unable to check convergence. More?
3
1
1 1 n+3
IfBReHb eL > 0 л ReJ ееееее N < ееееее , ееееее 2 еееееnееееее p Hb eL ееее2еnе ее s3
n
3 3
3
3 1
3 3
JGJ1 - ееееееееее N J1 F1 J1 - ееееееееее ; ееееее ; b eN - 2 b e 1 F1 J1 - ееееееееее ; еееее ; b eNN +
2n
2n 2
2n 2
3 Hn - 1L 3
3 Hn - 1L
Х!!!!!!!!
еееееееееее ; еееее ; b eN b e J2 GJ ееееееееееееееее
ееееееееее N 1 F1 J ееееееееееееееее
2n
2
2n
n-3
n-3 1
GJ еееееееееееееееее N 1 F1 J еееееееееееееееее ; ееееее ; b eNNN N A ,
2n
2n 2
16 -4 Ht-1L t b e 1- ее3ее
8
IntegrateB еееееееее ?
p t n b e s3 N A - еееее ?-4 Ht-1L t b e p t-3Йn b e s3 N A ,
3
3
1
1
8t, 0, ╤<, Assumptions ь ReJ ееееее N ╔ ееееее н ReHb eL ╖ 0FF
n
3
The result shows that under the conditions ReH ееее1n L < ееее13 and ReHb eL > 0, the
integral exists and the SVC is represented by hypergeometric functions
1 F1 depending on the potential parameter n, the inverse temperature b,
and the potential depth e. If the conditions on n and b e are not satisfied,
we observe that the integral cannot be evaluated. A more usable
representation of the result for our further calculations is generated if we
supress the conditions under which the integral is solvable. We select
650
5.7 Second Virial Coefficient
Bc = Bc Й. a_. If@b_, c_, d___D > a c
3
1 n+3
еееее 2 еееееnееееее p Hb eL ееее2еnе е s3
3
3
3 1
3 3
JGJ1 - ееееееееее N J1 F1 J1 - ееееееееее ; ееееее ; b eN - 2 b e 1 F1 J1 - ееееееееее ; ееееее ; b eNN +
2n
2n 2
2n 2
3 Hn - 1L 3
3 Hn - 1L
Х!!!!!!!!
еееееееееее ; ееееее ; b eN b e J2 GJ ееееееееееееееее
еееееееееее N 1 F1 J ееееееееееееееее
2n
2
2n
n-3
n-3 1
GJ ееееееееееееее ее N 1 F1 J ееееееееееееее ее ; еееее ; b eNNN N A
2n
2n 2
The result is that the classical SVC for H2 n - nL potentials can be
represented by hypergeometric functions. A graphical representation of the
SVC in a scaled representation follows:
Plot@Evaluate@Bc Й. 8H ▒ 1, V ▒ 1, n ▒ 6, NA ▒ 1<D,
8E, 0., 1 Й 2<, AxesLabel ▒ 8"EH", "Bc ЙHNA V3 L"<,
PlotStyle ▒ RGBColor@0, 0, 0.996109DD;
Bc ЙHNA s3 L
1
0.5
0.1
0.2
0.3
0.4
0.5
be
-0.5
-1
The plot shows that the classical SVC possesses a single maximum in the
variable be. In addition to the graphical representation of the SVC, the
analytical result allows us to apply the result to thermodynamic quantities
as given in Eqs. (5.7.111-5.7.115). This opens the way to access
thermodynamic quantities like the internal erenrgy. The internal energy for
example is defined in terms of the SVC by
5. Quantum Mechanics
~
651
~0
B1
U -U
ееееееее
~ее + ?N
R Tееееее = -J ееее
V
which becomes
n
d B
Bn = T n ееее
ееееее ,
dTn
(5.7.118)
652
5.7 Second Virial Coefficient
1
≥IBcЙ.bф дддддддд
ддддддд M
k T
B
T дддддддддддддддд
дддддддддддддддд
ддддд
≥T
InternalEnergy = - дддддддддддддддддддддддддддддддд
ддддддддддддддддд
V
1
- ееееее
V
ij ij 1 n+3
jjT jj ееееее 2 еееееnееееее p s3
jj jj
3
k k
3 Hn - 1L 3
e
n-3
3 Hn - 1L
jij
jj-Je J2 GJ ееееееееееееееее
еееееееееее ; ееееее ; ееееееееееееее N - GJ еееееееееееееееее N
еееееееееее N 1 F1 J ееееееееееееееее
j
2n
2 T kB
2n
2n
k
yz
ij
e
n-3 1
e
ееееееееееееее kB zzz +
ееееее ; ееееееееееееее NNN Л jjj2 T 2 $%%%%%%%%%%%%%
T kB
2 n 2 T kB
{
k
1 F1 J еееееееееееееееее ;
3
3
e
ее е L F I2 - ееее
ее е ; ееее5 ; ееееееее
ееее M e2
3 ji 4 H1 - ееее
2n 1 1
2 n 2 T kB
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееееееее +
GJ1 - ееееееееее N jjjj ееееееееееееееееееееееееееееееее
2n
3 T 3 kB2
k
3
e
ее е ; ееее3 ; ееееееее
ееее M e
2 1 F1 I1 - ееее
2 n 2 T kB
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееееееее 2
T kB
3
3
e
ееее L F I2 - ееее
ееее ; ееее3 ; ееееееее
ееее M e yz
2 H1 - ееее
2n 1 1
2 n 2 T kB
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееее zzzz +
2
T kB
{
n-3
n-3
3
e
ееее M
jij Hn - 3L e GH ееее2ееееnее L 1 F1 I ееее2ееееnее + 1; ееее2 ; ееееееее
T kB
jj ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееее j
n T 2 kB
k
3 Hn-1L
3 Hn-1L
e
2 Hn - 1L e GH ееееееее
ееееееееее L 1 F1 I ееееееее
еееееееее + 1; ееее52 ; ееееееее
ееее M y
2n
2n
T kB z
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееее
ееееее zzzz
n T 2 kB
{
1
e yz e ееее2еnе е
еееееее
ееееееееееееее zzzz J ееееееееееееее N N A - ееееееееееееееее
$%%%%%%%%%%%%%
n T 2 kB
T kB
T kB
{
3
3 Hn - 1L
3 Hn - 1L 3
e
ji
jij ееееn+3
jj2 еnееееее -1 p e s3 jjj$%%%%%%%%%%%%%
еееееееееее N 1 F1 J ееееееееееееееее
еееееееееее ; ееееее ;
ееееее J2 GJ ееееееееееееееее
j ееееееее
j
2n
2n
2
T kB
k
k
e
n-3
n-3 1
e
ееееееееееееее N - GJ еееееееееееееееее N 1 F1 J еееееееееееееееее ; ееееее ; ееееееееееееее NN +
T kB
2n
2 n 2 T kB
3 ij
3 1
e
GJ1 - ееееееееее N jjjj1 F1 J1 - ееееееееее ; ееееее ; ееееееееееееее N 2n
2 n 2 T kB
k
3
e
2 e 1 F1 I1 - ееее
ееее ; ееее3 ; ееееееее
еее M yyz e ееее23еnе е -1 yzyzyz
2 n 2 T kB z
N A zzzzzzzzzzzz
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееееееее zzzzzzzz J ееееееееееееее N
T kB
T kB
{{
{{{
In the above line, we used relation (5.7.111) to represent the internal
energy. Since the SVC in our calculations does not depend explicitly on
5. Quantum Mechanics
653
the temperature T, we replaced the reduced temperature b by 1 Й HkB TL.
After this replacement in Bc , we differentiate the resulting expression with
respect to T. A multiplication of the result by T and a normalization with
the volume V delivers the final result. All of these steps are contained in
the above input line. The result is a general analytic expression for the
internal energy allowing the choice of the temperature T, the potential
depth e, the radius s, and the exponent of the potential n. To describe a
specific gas, we have to insert numeric values for the parameters into the
result.
For
example,
we
find
for
e = 1, n = 6, s = 1., NA = 1, kB = 1, T = 200, and V = 1 an internal energy
of
<<Miscellaneous`PhysicalConstants`
InternalEnergy Й.
8e ф 10-20 Joule , n ф 6, s ф 10-8 , N A ф AvogadroConstant,
k B ф BoltzmannConstant, T ф 200 Kelvin, V ф 1<
132.423
- ееееееееееееееее
ееееееееее
Mole
By inserting the model parameters e, n, s, and the other thermodynamic
parameters NA , kB , and V , we have access to the numerical values of the
internal energy as well. If we vary the temperature T, these values show
the dependence of the internal energy on T. If we are interested in the
temperature dependence of the internal energy, we can generate a plot by
654
5.7 Second Virial Coefficient
Plot@Evaluate@InternalEnergy Mole Й. 8H ▒ 1020 Joule,
n ▒ 6, V ▒ 108 , NA ▒ AvogadroConstant,
kB ▒ BoltzmannConstant, T ▒ t Kelvin, V ▒ 1<D,
8t, 200, 250<, AxesLabel ▒ 8"T", "u"<,
PlotStyle ▒ RGBColor@0.996109, 0, 0DD;
u
-60
-70
-80
-100
-110
-120
-130
210
220
230
240
250
T
If we change, in addition to T, the exponent n in the potential, we get the
following figure.
5. Quantum Mechanics
655
Plot@Evaluate@Table@
InternalEnergy Mole Й. 8H ▒ 1020 Joule, n ▒ Q, V ▒ 108 ,
NA ▒ AvogadroConstant, kB ▒ BoltzmannConstant,
T ▒ t Kelvin, V ▒ 1<, 8Q, 4, 9, 1<DD,
8t, 200, 250<, AxesLabel ▒ 8"T", "u"<,
PlotStyle ▒ RGBColor@0.996109, 0, 0D,
Prolog ▒ 8Text@"n=4", 8224.474, 199.729<D,
Text@"n=8", 8224.474, 40<D,
Text@"n=5", 8224.474, 130<D<D;
u
210
-50
220
230
n=8
240
250
T
-100
-150
-200
n=5
n=4
-250
The reader can determine other thermodynamic quantities of his interest,
such as enthalpy, entropy, heat capacity at constant pressure, or the
Joul?Thomson coefficient.
5.7.3 Quantum Mechanical Corrections Bq1 HTL and Bq2 HTL of
the SVC
Up to the present considerations, we only know the classical behavior of
the gas for high temperatures. The following discussion includes two
quantum mechanical corrections allowing us to discuss all thermodynamic
quantities in cases where quantum corrections are necessary.
The quantum mechanical corrections Bq1 and Bq2 in Eq. (5.7.109) and
(5.7.110) are realized by the same substitution as demonstrated in the
656
5.7 Second Virial Coefficient
classical calculation. The integrand of the first quantum correction is
transformed by
integrandQc1 =
≥UHrL 2
2 p N A b3 ?-b UHrL r2 I дддддддд
дддддд M DifferentialDHrL
≥r
SimplifyA дддддддддддддддддддддддддддддддд
дддддддддддддддддддддддддддддддд
дддддддддддддддддддддддддддддддд
дддддддддддддддддддддддддддддддд
дддддддддддддд Й. substitution Й.
48 p2
DifferentialDHtL ф 1E
n
n
i 1 y ii 1 y
y
-4 jjt ееnеее zz jjjjt ееnеее zz -1zz b e
2 ? k { kk { {
n+1
1
2n
1
n 2
n t- еееееnееееее It ееnее M I1 - 2 It ееnее M M b3 e2 s N A
- ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееее
3p
The related integral follows by inserting the integrand into the integral:
Bq1 = ?
?
integrandQc1 ? t
0
IfAReHb eL > 0,
1
1 1
1
1
1 1
- ееееееееее J2 ееnее -2 n b3 e2 Hb eL ее2ее I ееnее -5M s JGJ1 - ееееееееее N 1 F1 J1 - ееееееееее ; ееееее ; b eN Hb eL3Й2 +
3p
2n
2n 2
1
1 1
1 3
GJ2 - ееееееееее N J1 F1 J2 - ееееееееее ; ееееее ; b eN - 4 b e 1 F1 J2 - ееееееееее ; ееееее ; b eNN
2n
2n 2
2n 2
3
1
Х!!!!!!!!
b e + 2 b e JGJ еееее - ееееееееее N
2
2n
3
1 3
3
1 1
Jb e 1 F1 J ееееее - ееееееееее ; еееее ; b eN - 1 F1 J еееее - ееееееееее ; ееееее ; b eNN +
2
2n 2
2
2n 2
5
1
5
1 3
GJ ееееее - ееееееееее N 1 F1 J ееееее - ееееееееее ; ееееее ; b eNNN N A N,
2
2n
2
2n 2
n
n
i 1 y ii 1 y
y
-4 jjt ееnеее zz jjjjt ееnеее zz -1zz b e
n+1
1
2n
2 ? k { kk { { n t- еееееnееееее b3 e2 s N A It ееnее M
IntegrateA- ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееееее +
3p
n
n
i 1 y ii 1 y
y
-4 jjt ееnеее zz jjjjt ееnеее zz -1zz b e
n+1
1
3n
n+1
1
4n
8 ? k { kk { { n t- еееееnееееее b3 e2 s N A It ееnее M
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееееее 3p
n
n
i 1 y ii 1 y
y
-4 jjt ееnеее zz jjjjt ееnеее zz -1zz b e
8 ? k { kk { {
n t- еееееnееееее b3 e2 s N A It ееnее M
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееееее ,
3p
8t, 0, ╤<, Assumptions ь ReHb eL ╖ 0EE
5. Quantum Mechanics
657
Bq1 = Bq1 Й. a_. If@b_, c_, d___D > a c
1
1 1
1
- ееееееееее J2 ееnее -2 n b3 e2 Hb eL ее2ее I ееnее -5M s
3p
1
1 1
1
JGJ1 - ееееееееее N 1 F1 J1 - ееееееееее ; ееееее ; b eN Hb eL3Й2 + GJ2 - ееееееееее N
2n
2n 2
2n
1 1
1 3
Х!!!!!!!!
J1 F1 J2 - ееееееееее ; ееееее ; b eN - 4 b e 1 F1 J2 - ееееееееее ; ееееее ; b eNN b e +
2n 2
2n 2
3
1
3
1 3
3
1
2 b e JGJ еееее - ееееееееее N Jb e 1 F1 J еееее - ееееееееее ; ееееее ; b eN - 1 F1 J ееееее - ееееееееее ;
2
2n
2
2n 2
2
2n
5
1 3
1
5
1
ееееее ; b eNN + GJ ееееее - ееееееееее N 1 F1 J ееееее - ееееееееее ; ееееее ; b eNNN N A N
2
2n 2
2
2
2n
Again, we find an analytic representation of the first quantum mechanical
correction of the SVC by means of hypergeometric functions 1 F1 . The
integrand for the second quantum correction Bq2 follows by
integrandQc2 = SimplifyA
1
PowerExpandA- дддддддддддддддд
ддддддддд
1920 p4
ij
ij
≥UHrL 4
jj
j 1
jj2 p N A b4 ?-b UHrL jjj- дддддддддд 5 b2 jij ддддддддддддддддддддд zyz +
jjj
jjj 36
k ≥r {
k
k
2y
10 b I дддддддд
дддддд M
дддддд M
2 I дддддддд
i ≥2 UHrL yz zzz 2
≥r
≥r
дддддддд zzz zzz r
дддддддддддддддд
дддддддддддддддд
дддддддддд + дддддддддддддддд
ддддддддддддддддд + jjjj дддддддддддддддд
z
9r
r
k ≥r ≥r { z{
≥UHrL 3
≥UHrL 2
zyz
DifferentialDHrLzzzz Й. substitution Й. DifferentialDHtL ф 1EE
zz
{
1
1
- ееееееееееееееее
ееееееееееее I?-4 Ht-1L t b e n t b4 e2 I4 H18 n2 + H27 - 10 b eL n + 9L t1+ ееnее +
540 p3 s
1
4 HH5 b2 e2 - 36L n2 + 12 H5 b e - 3L n - 9L t2+ ееnее - 160 n b e
1
1
1
Hn b e + 3L t3+ ееnее + 160 n b e H3 n b e + 2L t4+ ееnее - 640 n2 b2 e2 t5+ ееnее +
1
1
320 n2 b2 e2 t6+ ееnее - 9 Hn + 1L2 t ееnее - 72 s t2 + 72 s t - 18 sM N A M
and the explicit integration provides
658
5.7 Second Virial Coefficient
Bq2 = ?
?
integrandQc2 ? t
0
Integrate::gener : Unable to check convergence. More?
1
- ееееееееееееееее
ееееееееееее
540 p3 s
ij
1
jj 4 2
jjn b e IfBReJ ееееее N > -2 л ReHb eL > 0,
jj
n
k
1
1
1
1
5 2-1- ееnее n GH ееее12 H3 + ееее1n LL 1 F1 H ееее32 + ееее
ееее ; ееее1 ; b eL Hb eL ее2ее - ееее2еnе е
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееее +
- ееееееееееееееееееееееееееееееее
be
1
1
1
1
ееее ; ееее1 ; b eL Hb eL ее2ее - ееее2еnе е
9 2-2- ееnее GH ее12ее H3 + ееее1n LL 1 F1 H ее32ее + ееее
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееееееее +
b2 e2
1
1
1
1
1
1
ееее ; ееее1 ; b eL Hb eL ее2ее - ееее2еnе е
9 2-1- ееnее n2 GH ееее12 H3 + ееее1n LL 1 F1 H ее32ее + ееее
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееееееее +
b2 e2
1
1
ееее ; ее1ее ; b eL Hb eL ее2ее - ееее2еnе е
27 2-2- ееnее n GH ееее12 H3 + ееее1n LL 1 F1 H ееее32 + ееее
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееееееее b2 e2
1
1
1
1
ееее ; ееее3 ; b eL Hb eL ее2ее - ееее2еnе е
9 2-2- ееnее GH ее12ее H3 + ееее1n LL 1 F1 H ее32ее + ееее
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееееееее be
1
1
1
1
ееее ; ееее3 ; b eL Hb eL ее2ее - ееее2еnе е
9 2-2- ееnее n2 GH ееее12 H3 + ееее1n LL 1 F1 H ее32ее + ееее
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееееееее be
1
1
1
1
ееее ; ее3ее ; b eL Hb eL ее2ее - ееее2еnе е
9 2-1- ееnее n GH ееее12 H3 + ееее1n LL 1 F1 H ееее32 + ееее
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееее +
be
1
1
1
1
1
5
1 3
5 2-2- ееnее n2 GJ ееееее J5 + ееееее NN 1 F1 J ееееее + ееееееееее ; ееееее ; b eN Hb eL ее2ее - ееее2еnе е n
2
2
2n 2
1
3
1
3
1
1
5 2-1- ееnее n2 GH ееее12 H5 + ееее1n LL 1 F1 H ее52ее + ееее
ееее ; ееее1 ; b eL Hb eL ее2ее - ееее2еnе е
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееееееее 2
2
b e
1
1
ееее ; ее1ее ; b eL Hb eL ее2ее - ееее2еnе е
15 2-1- ееnее n GH ееее12 H5 + ееее1n LL 1 F1 H ееее52 + ееее
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееееееее +
3
3
b e
3
1
1
ееее ; ее3ее ; b eL Hb eL ее2ее - ееее2еnе е
15 2-1Йn n GH ееее12 H5 + ееее1n LL 1 F1 H ееее52 + ееее
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееее b2 e2
1
3
1
1
ееее ; ееее3 ; b eL Hb eL ее2ее - ееее2еnе е
9 2-2- ееnее GH ее12ее H5 + ееее1n LL 1 F1 H ее52ее + ееее
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееееееее b3 e3
3
1
1
ееее ; ееее3 ; b eL Hb eL ее2ее - ееее2еnе е
9 2-1Йn n2 GH ееее12 H5 + ееее1n LL 1 F1 H ее52ее + ееее
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееее b3 e3
5. Quantum Mechanics
659
3
1
1
9 2-1Йn n GH ееее12 H5 + ееее1n LL 1 F1 H ееее52 + ееее
ееее ; ее3ее ; b eL Hb eL ее2ее - ееее2еnе е
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееееееее b3 e3
5
1
1
1
5 2-1- ееnее n2 GH ееее12 H7 + ееее1n LL 1 F1 H ее72ее + ееее
ееее ; ееее1 ; b eL Hb eL ее2ее - ееее2еnе е
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееееееее +
4
4
b e
5
1
1
1
15 2-1- ееnее n2 GH ее12ее H7 + ееее1n LL 1 F1 H ее72ее + ееее
ееее ; ееее3 ; b eL Hb eL ее2ее - ееее2еnе е
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее +
3
3
b e
5
1
1
5 2-1Йn n GH ееее12 H7 + ееее1n LL 1 F1 H ееее72 + ееее
ееее ; ее3ее ; b eL Hb eL ее2ее - ееее2еnе е
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееееееее +
4
4
b e
7
1
1
1
5 2-2- ееnее n2 GH ееее12 H9 + ееее1n LL 1 F1 H ее92ее + ееее
ееее ; ееее3 ; b eL Hb eL ее2ее - ееее2еnе е
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееееееее 5
5
b e
1
1
1
1
9 2-3- ееnее GH1 + ееее
ееее L F H1 + ееее
ееее ; ееее1 ; b eL Hb eL- ееее2еnе ее
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееее be
1
1
1
1
9 2-3- ееnее GH ееее
ееее L F H1 + ееее
ееее ; ееее1 ; b eL Hb eL- ееее2еnе е
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееее be
1
1
1
1
ее е L F H1 + ееее
ее е ; ееее1 ; b eL Hb eL- ееее2еnе е
9 2-4- ееnее n GH ееее
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееееееее +
be
1
1
1
1 1
5 2-3- ееnее n2 GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; ееееее ; b eN Hb eL- ееее2еnе е +
2n
2n 2
1
1
1
1
15 2-1- ееnее n GH2 + ееее
ееее L F H2 + ееее
ееее ; ее1ее ; b eL Hb eL- ееее2еnе е
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееее be
1
1
1
1
9 2-3- ееnее GH2 + ееее
ееее L F H2 + ееее
ееее ; ееее1 ; b eL Hb eL- ееее2еnе ее
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееее b2 e2
1
1
1
1
9 2-1- ееnее n2 GH2 + ееее
ееее L F H2 + ееее
ееее ; ееее1 ; b eL Hb eL- ееее2еnе ее
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееее b2 e2
1
1
1
1
9 2-1- ееnее n GH2 + ееее
ееее L F H2 + ееее
ееее ; ее1ее ; b eL Hb eL- ееее2еnе е
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееее b2 e2
1
1
1 3
5 2-1Йn n GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; ееееее ; b eN Hb eL- ееее2еnе е +
2n
2n 2
1
1
1
1
9 2-1- ееnее GH2 + ееее
ееее L F H2 + ееее
ееее ; ееее3 ; b eL Hb eL- ееее2еnе ее
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееее +
be
1
1
1
9 2-1Йn n2 GH2 + ееее
ееее L F H2 + ееее
ееее ; ееее3 ; b eL Hb eL- ееее2еnе ее
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееее +
be
1
1
1
1
27 2-1- ееnее n GH2 + ееее
ееее L F H2 + ееее
ееее ; ее3ее ; b eL Hb eL- ееее2еnе е
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееее +
be
660
5.7 Second Virial Coefficient
1
1
1
1
15 2-2- ееnее n2 GH3 + ееее
ееее L F H3 + ееее
ееее ; ееее1 ; b eL Hb eL- ееее2еnе ее
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееее +
be
1
1
1
1
5 2-1- ееnее n GH3 + ееее
ееее L F H3 + ееее
ееее ; ее1ее ; b eL Hb eL- ееее2еnе е
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееее 2
2
b e
1
1
1 3
5 2-1Йn n2 GJ3 + ееееееееее N 1 F1 J3 + ееееееееее ; ееееее ; b eN Hb eL- ееее2еnе е 2n
2n 2
1
1
1
15 2-1Йn n GH3 + ееее
ееее L F H3 + ееее
ееее ; ее3ее ; b eL Hb eL- ееее2еnе е
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееее +
be
1
1
1
1
ееее L F H4 + ееее
ееее ; ееее1 ; b eL Hb eL- ееее2еnе ее
5 2-3- ееnее n2 GH4 + ееее
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееее b2 e2
1
1
1
5 2-1Йn n2 GH4 + ееее
ееее L F H4 + ееее
ееее ; ееее3 ; b eL Hb eL- ееее2еnе ее
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееее be
Х!!!! 2 b e
27 ? b e p I ееееееее
еее + 1M s Hb eL5Й2
3
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееее +
8 b4 e4
Х!!!!
Х!!!!
9 ? b e p s Hb eL3Й2
9 ? b e p H2 b e + 1L s Hb eL3Й2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееее
ееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееее +
4 b3 e3
4 b2 e2
Х!!!!
Х!!!!!!!!
Х!!!! Х!!!!!!!!
2 ?-b e
9 ? b e s I2 p I1 - erfI b e MM - ееееееее
ееееееее - 2 p M b e
Х!!!!!!!!
be
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееееееее +
8 be
1
ееееееееееееееее
2 be
ij
i
! Х!!!! Х!!!!
Х!!!!!!!!
jj9 s jjj? b e Х!!!!!!!
b e I p - p I1 - erfI b e MMM +
j
j
k
k
Х!!!! Х!!!!
Х!!!!!!!!
? b e I p - p I1 - erfI b e MMM
zyzy
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееее + 1zzzzzz Х!!!!!!!! ееееееееееееееее
2 be
{{
1
ееееееееееееееее
еееее
8 b2 e2
2 ?-b e
Х!!!! y
Х!!!!!!!!
jij9 s jij-? b e jij2 Х!!!!
ееееееееееее! е - 2 p zzzz
p I1 - erfI b e MM - ееееееее
jj
jj
jj
Х!!!!!!!
b
e
{
k
k
k
3 be
3Й2
Hb eL - ееееее ?
2
yy
ij Х!!!!
2 ?-b e
Х!!!! y Х!!!!!!!!
!
jjj2 p I1 - erfIХ!!!!!!!
ееееееееееее! е - 2 p zzzz b e - 1zzzzzzzz,
b e MM - ееееееее
Х!!!!!!!
be
{
{{
k
1
IntegrateB?-4 Ht-1L t b e t I4 H18 n2 + H27 - 10 b eL n + 9L t1+ ееnее +
1
4 HH5 b2 e2 - 36L n2 + 12 H5 b e - 3L n - 9L t2+ ееnее 1
1
160 n b e Hn b e + 3L t3+ ееnее + 160 n b e H3 n b e + 2L t4+ ееnее 640 n2 b2 e2 t
ее1nее
5+ ее1nее
+ 320 n2 b2 e2 t
6+ ее1nее
-
9 Hn + 1L t - 72 s t + 72 s t - 18 sM, 8t, 0, ╤<,
2
2
yz
1
z
Assumptions ь ? JReJ ееееее N > -2 л ReHb eL > 0NFF N A zzzz
n
z
{
5. Quantum Mechanics
661
Bq2 = Bq2 Й. a_. If@b_, c_, d___D ф a c
1
- ееееееееееееееее
ееееееееееее
540 p3 s
1
1
ij
i
1
1
3
1
-1- ее1ее
еnеее ; ееее12 ; b eL Hb eL ее2ее - ееее2еnе е
jj 4 2 jjj 5 2 n n GH ееее2 H3 + ееееn LL 1 F1 H ееее2 + ееее
2
jjn b e jj- ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееее е +
jj
jj
be
k
k
1
1
1
1
9 2-2- ееnее GH ее12ее H3 + ееее1n LL 1 F1 H ее32ее + ееее
ееее ; ееее1 ; b eL Hb eL ее2ее - ееее2еnе е
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееееееее +
b2 e2
1
1
1
1
1
1
ееее ; ееее1 ; b eL Hb eL ее2ее - ееее2еnе е
9 2-1- ееnее n2 GH ееее12 H3 + ееее1n LL 1 F1 H ее32ее + ееее
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееееееее +
2
2
b e
1
1
ееее ; ее1ее ; b eL Hb eL ее2ее - ееее2еnе е
27 2-2- ееnее n GH ееее12 H3 + ееее1n LL 1 F1 H ееее32 + ееее
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееееееее 2
2
b e
1
1
1
1
ееее ; ееее3 ; b eL Hb eL ее2ее - ееее2еnе е
9 2-2- ееnее GH ее12ее H3 + ееее1n LL 1 F1 H ее32ее + ееее
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееееееее be
1
1
1
1
9 2-2- ееnее n2 GH ееее12 H3 + ееее1n LL 1 F1 H ее32ее + ееее
ееее ; ееее3 ; b eL Hb eL ее2ее - ееее2еnе е
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееееееее be
1
1
1
1
9 2-1- ееnее n GH ееее12 H3 + ееее1n LL 1 F1 H ееее32 + ееее
ееее ; ее3ее ; b eL Hb eL ее2ее - ееее2еnе е
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееее +
be
1
1
1
1
1
5
1 3
5 2-2- ееnее n2 GJ ееееее J5 + ееееее NN 1 F1 J ееееее + ееееееееее ; ееееее ; b eN Hb eL ее2ее - ееее2еnе е n
2
2
2n 2
1
3
1
3
1
1
5 2-1- ееnее n2 GH ееее12 H5 + ееее1n LL 1 F1 H ее52ее + ееее
ееее ; ееее1 ; b eL Hb eL ее2ее - ееее2еnе е
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееееееее b2 e2
1
1
ееее ; ее1ее ; b eL Hb eL ее2ее - ееее2еnе е
15 2-1- ееnее n GH ееее12 H5 + ееее1n LL 1 F1 H ееее52 + ееее
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееееееее +
b3 e3
3
1
1
ееее ; ее3ее ; b eL Hb eL ее2ее - ееее2еnе е
15 2-1Йn n GH ееее12 H5 + ееее1n LL 1 F1 H ееее52 + ееее
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееее b2 e2
1
3
1
1
ееее ; ееее3 ; b eL Hb eL ее2ее - ееее2еnе е
9 2-2- ееnее GH ее12ее H5 + ееее1n LL 1 F1 H ее52ее + ееее
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееееееее b3 e3
3
1
1
ееее ; ееее3 ; b eL Hb eL ее2ее - ееее2еnе е
9 2-1Йn n2 GH ееее12 H5 + ееее1n LL 1 F1 H ее52ее + ееее
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееее b3 e3
3
1
1
9 2-1Йn n GH ееее12 H5 + ееее1n LL 1 F1 H ееее52 + ееее
ееее ; ее3ее ; b eL Hb eL ее2ее - ееее2еnе е
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееееееее b3 e3
662
5.7 Second Virial Coefficient
5
1
1
1
5 2-1- ееnее n2 GH ееее12 H7 + ееее1n LL 1 F1 H ее72ее + ееее
ееее ; ееее1 ; b eL Hb eL ее2ее - ееее2еnе е
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееееееее +
b4 e4
5
1
1
1
15 2-1- ееnее n2 GH ее12ее H7 + ееее1n LL 1 F1 H ее72ее + ееее
ееее ; ееее3 ; b eL Hb eL ее2ее - ееее2еnе е
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее +
3
3
b e
5
1
1
5 2-1Йn n GH ееее12 H7 + ееее1n LL 1 F1 H ееее72 + ееее
ееее ; ее3ее ; b eL Hb eL ее2ее - ееее2еnе е
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееееееее +
4
4
b e
7
1
1
1
5 2-2- ееnее n2 GH ееее12 H9 + ееее1n LL 1 F1 H ее92ее + ееее
ееее ; ееее3 ; b eL Hb eL ее2ее - ееее2еnе е
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееееееее 5
5
b e
1
1
1
1
9 2-3- ееnее GH1 + ееее
ееее L F H1 + ееее
ееее ; ееее1 ; b eL Hb eL- ееее2еnе ее
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееее be
1
1
1
1
9 2-3- ееnее GH ееее
ееее L F H1 + ееее
ееее ; ееее1 ; b eL Hb eL- ееее2еnе е
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееее be
1
1
1
1
9 2-4- ееnее n GH ееее
ее е L F H1 + ееее
ее е ; ееее1 ; b eL Hb eL- ееее2еnе е
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееееееее +
be
1
1
1
1 1
5 2-3- ееnее n2 GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; ееееее ; b eN Hb eL- ееее2еnе е +
2n
2n 2
1
1
1
1
ееее L F H2 + ееее
ееее ; ее1ее ; b eL Hb eL- ееее2еnе е
15 2-1- ееnее n GH2 + ееее
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееее be
1
1
1
1
9 2-3- ееnее GH2 + ееее
ееее L F H2 + ееее
ееее ; ееее1 ; b eL Hb eL- ееее2еnе ее
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееее b2 e2
1
1
1
1
9 2-1- ееnее n2 GH2 + ееее
ееее L F H2 + ееее
ееее ; ееее1 ; b eL Hb eL- ееее2еnе ее
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееее b2 e2
1
1
1
1
9 2-1- ееnее n GH2 + ееее
ееее L F H2 + ееее
ееее ; ее1ее ; b eL Hb eL- ееее2еnе е
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееее b2 e2
1
1
1 3
5 2-1Йn n GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; ееееее ; b eN Hb eL- ееее2еnе е +
2n
2n 2
1
1
1
1
9 2-1- ееnее GH2 + ееее
ееее L F H2 + ееее
ееее ; ееее3 ; b eL Hb eL- ееее2еnе ее
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееее +
be
1
1
1
9 2-1Йn n2 GH2 + ееее
ееее L F H2 + ееее
ееее ; ееее3 ; b eL Hb eL- ееее2еnе ее
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееее +
be
1
1
1
1
27 2-1- ееnее n GH2 + ееее
ееее L F H2 + ееее
ееее ; ее3ее ; b eL Hb eL- ееее2еnе е
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееее +
be
1
1
1
1
15 2-2- ееnее n2 GH3 + ееее
ееее L F H3 + ееее
ееее ; ееее1 ; b eL Hb eL- ееее2еnе ее
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееее +
be
5. Quantum Mechanics
663
1
1
1
1
5 2-1- ееnее n GH3 + ееее
ееее L F H3 + ееее
ееее ; ее1ее ; b eL Hb eL- ееее2еnе е
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееее b2 e2
1
1
1 3
5 2-1Йn n2 GJ3 + ееееееееее N 1 F1 J3 + ееееееееее ; ееееее ; b eN Hb eL- ееее2еnе е 2n
2n 2
1
1
1
15 2-1Йn n GH3 + ееее
ееее L F H3 + ееее
ееее ; ее3ее ; b eL Hb eL- ееее2еnе е
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееее +
be
1
1
1
1
ееее L F H4 + ееее
ееее ; ееее1 ; b eL Hb eL- ееее2еnе ее
5 2-3- ееnее n2 GH4 + ееее
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееее b2 e2
1
1
1
ееее L F H4 + ееее
ееее ; ееее3 ; b eL Hb eL- ееее2еnе ее
5 2-1Йn n2 GH4 + ееее
2n 1 1
2n 2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееее be
Х!!!! 2 b e
27 ? b e p I ееееееее
еее + 1M s Hb eL5Й2
3
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееее +
8 b4 e4
Х!!!!
Х!!!!
9 ? b e p s Hb eL3Й2
9 ? b e p H2 b e + 1L s Hb eL3Й2
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееее
ееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееее +
4 b3 e3
4 b2 e2
Х!!!!
Х!!!!!!!!
Х!!!! Х!!!!!!!!
2 ?-b e
9 ? b e s I2 p I1 - erfI b e MM - ееееееее
ееееееее - 2 p M b e
Х!!!!!!!!
be
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееееееее +
8 be
1
ееееееееееееееее
2 be
ij
i
! Х!!!! Х!!!!
Х!!!!!!!!
jj9 s jjj? b e Х!!!!!!!
b e I p - p I1 - erfI b e MMM +
j
j
k
k
Х!!!! Х!!!!
Х!!!!!!!!
? b e I p - p I1 - erfI b e MMM
1
zyzy
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееее + 1zzzzzz - ееееееееееееееее
еееее
Х!!!!!!!! ееееееееееееееее
2 e2
8
b
2 be
{{
2 ?-b e
Х!!!! y
Х!!!!!!!!
jij9 s jij-? b e jij2 Х!!!!
ееееееееее!ее - 2 p zzzz Hb eL3Й2 p I1 - erfI b e MM - ееееееее
jj
jj
jj
Х!!!!!!!
b
e
{
k
k
k
i Х!!!!
3
2 ?-b e
Х!!!! y Х!!!!!!!!
Х!!!!!!!!
ееееееееееее! е - 2 p zzzz b e ееееее ? b e jjjj2 p I1 - erfI b e MM - ееееееее
Х!!!!!!!
2
be
{
k
y y
yyzz zz
1zzzzzzzzzzzz N A zzzz
{{z{ z{
Using e Й kB and ееее23 p NA s3 in a scaling transformation for the temperature
in the SVC, we get B* HTL tbulated in books like Hirschfelder et al [5.10].
These authors introduce a scaled representation of the SVC by
B*c = Bc Й H ееее23 p N A s3 L with a reduced temperature of T * = kB T Й e.
664
5.7 Second Virial Coefficient
я2 2
Bq1 я2
Bq2 I ддддmддд M + ддддддддmдддддддддд + Bc
1
Х!!!!!!!!!
BStar = SimplifyA дддддддддддддддддддддддддддддддд
дддддддддддддддддддддддддддддддд
ддддддддддддддд ЙЙ. 9e ф ддддддддддддд , я ф L s m e =E
2
3
b
T
дд3дд p N A s
1
ееееееееееееееее4е
45 p
1
ij
jj -7- ее1nее 1 4- ееее2еnе е
jj2
J ееееее N
jj
T
k
1
1 1
ij
1 ееее2еnе е -2
1 ее2ее I ееnее -3M
jj 2+ ее1nее
4
1+ ее1nее ееT1еее Х!!!! 4
jj9 2
n L s J ееееее N
+92
? n p L s J ееееее N
+
jj
T
T
k
1
1
9 21+ ееnее ? ееTеее n
ij 1 yz 1 ее12ее I ее1nее -3M
Х!!!! 4
p L s erf jjjj$%%%%%%%
ееееее % zzzz J ееееее N
T
T
k
{
1
2
1
1 1 1 1 ееnее -3
45 24+ ееnее n p2 L2 GJ2 - ееееееееее N 1 F1 J2 - ееееееееее ; еееее ; ееееее N J ееееее N
+
2n
2n 2 T T
1
5
1
5
2
3
1
3
1 1 1 1 ееnее - ее2ее
45 25+ ееnее n p2 L2 GJ ееееее - ееееееееее N 1 F1 J ееееее - ееееееееее ; еееее ; ееееее N J ееееее N
2
2n
2
2n 2 T T
2
5
1
5
1 3 1 1 ееnее - ее2ее
45 25+ ееnее n p2 L2 GJ ееееее - ееееееееее N 1 F1 J ееееее - ееееееееее ; еееее ; ееееее N J ееееее N
2
2n
2
2n 2 T T
1
2
1
1 1 1 1 ееnее -2
+
45 24+ ееnее n p2 L2 GJ1 - ееееееееее N 1 F1 J1 - ееееееееее ; еееее ; ееееее N J ееееее N
2n
2n 2 T T
1
2
1
1 3 1 1 ееnее -2
45 26+ ееnее n p2 L2 GJ2 - ееееееееее N 1 F1 J2 - ееееееееее ; еееее ; ееееее N J ееееее N
2n
2n 2 T T
1
3
2
3
1
3
1 3 1 1 ееnее - ее2ее
45 25+ ееnее n p2 L2 GJ ееееее - ееееееееее N 1 F1 J ееееее - ееееееееее ; еееее ; ееееее N J ееееее N
2
2n
2
2n 2 T T
2
7
4
n-3
n - 3 1 1 1 ееnее - ее2ее
+
45 27+ ееnее p4 GJ ееееееееееееее ее N 1 F1 J ееееееееееееее ее ; еееее ; ееееее N J ееееее N
2n
2n 2 T T
2
7
4
4
3 Hn - 1L
3 Hn - 1L 3 1
1 ееnее - ее2ее
45 28+ ееnее p4 GJ ееееееееееееееее
еееееееееее N 1 F1 J ееееееееееееееее
еееееееееее ; ееееее ; ееееее N J ееееее N
+ 45 27+ ееnее
2n
2n
2 T T
3
3 1 1
3 3 1
4
p GJ1 - ееееееееее N JT 1 F1 J1 - ееееееееее ; ееееее ; ееееее N - 2 1 F1 J1 - ееееееееее ; ееееее ; ееееее NN
2n
2n 2 T
2n 2 T
2
1
1
1
1
1 3 1
1 ееnее -3
ееееее % +
- 20 n3 L4 GJ ееееее J5 + ееееее NN 1 F1 J еееее J5 + ееееее N; ееееее ; ееееее N $%%%%%%%
J ееееее N
T
n
n 2 T
T
2
2
1
1 1 1
18 n T L4 GJ1 + ееееееееее N 1 F1 J1 + ееееееееее ; еееее ; ееееее N +
2n
2n 2 T
1
1 1 1
9 n2 T L4 GJ ееееееееее N 1 F1 J1 + ееееееееее ; еееее ; ееееее N +
2n
2n 2 T
1
1 1 1
18 n T L4 GJ ееееееееее N 1 F1 J1 + ееееееееее ; ееееее ; ееееее N 2n
2n 2 T
5. Quantum Mechanics
665
1
1 1 1
10 n3 L4 GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; ееееее ; ееееее N +
2n
2n 2 T
1
1 1 1
72 n3 T 2 L4 GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; ееееее ; ееееее N +
2n
2n 2 T
1
1 1 1
72 n2 T 2 L4 GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; ееееее ; ееееее N +
2n
2n 2 T
1
1 1 1
18 n T 2 L4 GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; ееееее ; ееееее N 2n
2n 2 T
1
1 1 1
120 n2 T L4 GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; ееееее ; ееееее N +
2n
2n 2 T
1
1 3 1
80 n2 L4 GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; ееееее ; ееееее N 2n
2n 2 T
1
1 3 1
144 n3 T L4 GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; ееееее ; ееееее N 2n
2n 2 T
1
1 3 1
216 n2 T L4 GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; ееееее ; ееееее N 2n
2n 2 T
1
1 3 1
72 n T L4 GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; еееее ; ееееее N 2n
2n 2 T
1
1 1 1
40 n2 T 2 L4 GJ3 + ееееееееее N 1 F1 J3 + ееееееееее ; ееееее ; ееееее N 2n
2n 2 T
1
1 1 1
60 n3 T L4 GJ3 + ееееееееее N 1 F1 J3 + ееееееееее ; ееееее ; ееееее N +
2n
2n 2 T
1
1 3 1
80 n3 L4 GJ3 + ееееееееее N 1 F1 J3 + ееееееееее ; ееееее ; ееееее N +
2n
2n 2 T
1
1 3 1
240 n2 T L4 GJ3 + ееееееееее N 1 F1 J3 + ееееееееее ; ееееее ; ееееее N 2n
2n 2 T
1
1 1 1
10 n3 T 2 L4 GJ4 + ееееееееее N 1 F1 J4 + ееееееееее ; ееееее ; ееееее N +
2n
2n 2 T
1
1 3 1
80 n3 T L4 GJ4 + ееееееееее N 1 F1 J4 + ееееееееее ; ееееее ; ееееее N +
2n
2n 2 T
40 n2 L4 GH ееее12 H3 + ееее1n LL 1 F1 H ее12ее H3 + ееее1n L; ееее12 ; ееT1ее L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееееее +
"######
ееееT1е #
36 n3 L4 GH ееее12 H3 + ееее1n LL 1 F1 H ее12ее H3 + ееее1n L; ееее32 ; ееT1ее L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееееее +
"######
ееееT1е #
72 n2 L4 GH ееее12 H3 + ееее1n LL 1 F1 H ее12ее H3 + ееее1n L; ееее32 ; ееT1ее L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееееее +
"######
ееееT1е #
36 n L4 GH ееее12 H3 + ееее1n LL 1 F1 H ееее12 H3 + ееее1n L; ее32ее ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееее +
"######
ееее1е #
T
H5 +
H5 + ееее1n L; ееее12 ; ееT1ее L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееееее "######
ееееT1е #
40 n3 L4 GH ееее12
ееее1n LL 1 F1 H ее12ее
666
5.7 Second Virial Coefficient
240 n2 L4 GH ее12ее H5 + ееее1n LL 1 F1 H ее12ее H5 + ееее1n L; ееее32 ; ееT1ее L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееее "######
ееееT1е #
120 n3 L4 GH ее12ее H7 + ееее1n LL 1 F1 H ее12ее H7 + ееее1n L; ееее32 ; ееT1ее L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееее "######
ееееT1е #
72 n3 L4 GH ееее12 H3 + ееее1n LL 1 F1 H ее12ее H3 + ееее1n L; ееее12 ; ееT1ее L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееееее 3Й2
H ееееT1е L
108 n2 L4 GH ее12ее H3 + ееее1n LL 1 F1 H ее12ее H3 + ееее1n L; ееее12 ; ееT1ее L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееее 3Й2
H ееT1ее L
36 n L4 GH ееее12 H3 + ееее1n LL 1 F1 H ееее12 H3 + ееее1n L; ее12ее ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееее +
3Й2
H ееееT1е L
120 n2 L4 GH ее12ее H5 + ееее1n LL 1 F1 H ее12ее H5 + ееее1n L; ееее12 ; ееT1ее L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееее +
3Й2
H ееT1ее L
144 n3 L4 GH ее12ее H5 + ееее1n LL 1 F1 H ее12ее H5 + ееее1n L; ееее32 ; ееT1ее L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееее +
3Й2
H ееT1ее L
144 n2 L4 GH ее12ее H5 + ееее1n LL 1 F1 H ее12ее H5 + ееее1n L; ееее32 ; ееT1ее L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееее +
3Й2
H ееT1ее L
36 n L4 GH ееее12 H5 + ееее1n LL 1 F1 H ееее12 H5 + ееее1n L; ее32ее ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееее +
3Й2
H ееееT1е L
40 n3 L4 GH ееее12 H7 + ееее1n LL 1 F1 H ее12ее H7 + ееее1n L; ееее12 ; ееT1ее L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееееее 3Й2
H ееееT1е L
80 n2 L4 GH ееее12 H7 + ееее1n LL 1 F1 H ее12ее H7 + ееее1n L; ееее32 ; ееT1ее L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееееее 3Й2
H ееееT1е L
20 n3 L4 GH ееее12 H9 + ееее1n LL 1 F1 H ее12ее H9 + ееее1n L; ееее32 ; ееT1ее L yzzzyzzz
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееееее zzzzzz
3Й2
zz
H ееееT1е L
{{
where L = я Й HsHm eL1Й2 L is the reduced de Broglie wavelength of relative
motion. B* is an even polynomial of fourth order in L. It contains the
classical, first, and second quantum corrections as zeroth-, second-, and
fourth- order coefficients, respectively. We extract the reduced
representation of the second quantum correction by
5. Quantum Mechanics
667
bq2 = Coefficient@BStar, L, 4D
General::spell1 : Possible spelling error: new
symbol name "bq2" is similar to existing symbol "Bq2". More?
1
ееееееееееееееее4е
45 p
1
jij -7- ее1ее 1 4- ееее2еnе е
jj2 n J ееееее N
jj
T
j
k
1
1 1
ееее
jij 2+ ее1ее
2еnе е -2
1 ее2ее I ееnее -3M
1+ ее1nее ееT1еее Х!!!!
jj9 2 n n s J ееее1ее N
+
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ееее
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е
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p s erf jjjj$%%%%%%%
ееееее % z J ееееее N
j T zzz T
k
{
1
1
1
9 21+ ееnее ? ееTеее n
1
1
1
1
1
1 3 1
ееееее % +
20 n3 GJ еееее J5 + ееееее NN 1 F1 J еееее J5 + ееееее N; ееееее ; ееееее N $%%%%%%%
T
n
n 2 T
2
2
1
1 1 1
1
18 n T GJ1 + ееееееееее N 1 F1 J1 + ееееееееее ; ееееее ; ееееее N + 9 n2 T GJ ееееееееее N
2n
2n 2 T
2n
1 1 1
1
1 1 1
1 F1 J1 + ееееееееее ; ееееее ; ееееее N + 18 n T GJ ееееееееее N 1 F1 J1 + ееееееееее ; ееееее ; ееееее N 2n 2 T
2n
2n 2 T
1
1 1 1
1
10 n3 GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; ееееее ; ееееее N + 72 n3 T 2 GJ2 + ееееееееее N
2n
2n 2 T
2n
1 1 1
1
1 1 1
2 2
1 F1 J2 + ееееееееее ; ееееее ; ееееее N + 72 n T GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; ееееее ; ееееее N +
2n 2 T
2n
2n 2 T
1
1 1 1
1
18 n T 2 GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; еееее ; ееееее N - 120 n2 T GJ2 + ееееееееее N
2n
2n 2 T
2n
1 1 1
1
1 3 1
2
1 F1 J2 + ееееееееее ; ееееее ; ееееее N + 80 n GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; еееее ; ееееее N 2n 2 T
2n
2n 2 T
1
1 3 1
144 n3 T GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; еееее ; ееееее N 2n
2n 2 T
1
1 3 1
216 n2 T GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; еееее ; ееееее N 2n
2n 2 T
1
1 3 1
72 n T GJ2 + ееееееееее N 1 F1 J2 + ееееееееее ; ееееее ; ееееее N 2n
2n 2 T
1
1 1 1
40 n2 T 2 GJ3 + ееееееееее N 1 F1 J3 + ееееееееее ; ееееее ; ееееее N 2n
2n 2 T
1
1 1 1
3
60 n T GJ3 + ееееееееее N 1 F1 J3 + ееееееееее ; еееее ; ееееее N +
2n
2n 2 T
1
1
3 1
80 n3 GJ3 + ееееееееее N 1 F1 J3 + ееееееееее ; ееееее ; ееееее N +
2n
2n 2 T
1
1 3 1
240 n2 T GJ3 + ееееееееее N 1 F1 J3 + ееееееееее ; еееее ; ееееее N 2n
2n 2 T
1
1 1 1
10 n3 T 2 GJ4 + ееееееееее N 1 F1 J4 + ееееееееее ; ееееее ; ееееее N +
2n
2n 2 T
668
5.7 Second Virial Coefficient
1
1 3 1
80 n3 T GJ4 + ееееееееее N 1 F1 J4 + ееееееееее ; еееее ; ееееее N +
2n
2n 2 T
40 n2 GH ееее12 H3 + ееее1n LL 1 F1 H ееее12 H3 + ееее1n L; ееее12 ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееее +
"######
ееееT1е #
36 n3 GH ееее12 H3 + ееее1n LL 1 F1 H ееее12 H3 + ееее1n L; ееее32 ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееее +
"######
ееееT1е #
72 n2 GH ееее12 H3 + ееее1n LL 1 F1 H ееее12 H3 + ееее1n L; ееее32 ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееее +
"######
ееееT1е #
36 n GH ее12ее H3 + ее1nее LL 1 F1 H ееее12 H3 + ее1nее L; ееее32 ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееее +
"######
ееееT1е #
40 n3 GH ееее12 H5 + ееее1n LL 1 F1 H ееее12 H5 + ееее1n L; ееее12 ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееее "######
ееееT1е #
240 n2 GH ееее12 H5 + ееее1n LL 1 F1 H ееее12 H5 + ееее1n L; ееее32 ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееееееее "######
ееееT1е #
120 n3 GH ееее12 H7 + ееее1n LL 1 F1 H ееее12 H7 + ееее1n L; ееее32 ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееееееее "######
ееееT1е #
72 n3 GH ееее12 H3 + ееее1n LL 1 F1 H ееее12 H3 + ееее1n L; ееее12 ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееее 3Й2
H ееееT1е L
108 n2 GH ееее12 H3 + ееее1n LL 1 F1 H ееее12 H3 + ееее1n L; ееее12 ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееееееее 3Й2
H ееееT1е L
36 n GH ее12ее H3 + ее1nее LL 1 F1 H ееее12 H3 + ее1nее L; ееее12 ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееее +
3Й2
H ееT1ее L
120 n2 GH ееее12 H5 + ееее1n LL 1 F1 H ееее12 H5 + ееее1n L; ееее12 ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееееееее +
3Й2
H ееееT1е L
144 n3 GH ееее12 H5 + ееее1n LL 1 F1 H ееее12 H5 + ееее1n L; ееее32 ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееееееее +
3Й2
H ееееT1е L
144 n2 GH ееее12 H5 + ееее1n LL 1 F1 H ееее12 H5 + ееее1n L; ееее32 ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееееееее +
3Й2
H ееееT1е L
36 n GH ее12ее H5 + ее1nее LL 1 F1 H ееее12 H5 + ее1nее L; ееее32 ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееее +
3Й2
H ееT1ее L
5. Quantum Mechanics
669
40 n3 GH ееее12 H7 + ееее1n LL 1 F1 H ееее12 H7 + ееее1n L; ееее12 ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееее 3Й2
H ееееT1е L
80 n2 GH ееее12 H7 + ееее1n LL 1 F1 H ееее12 H7 + ееее1n L; ееее32 ; ееееT1е L
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееее 3Й2
H ееееT1е L
20 n3 GH ееее12 H9 + ееее1n LL 1 F1 H ееее12 H9 + ееее1n L; ееее32 ; ееееT1е L yzzzyzzz
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееее zzzzzz
3Й2
zz
H ееееT1е L
{{
The first quantum mechanical correction is extracted by
bq1 = Coefficient@BStar, L, 2D
General::spell1 : Possible spelling error: new
symbol name "bq1" is similar to existing symbol "Bq1". More?
1
ееееееееееееееее4е
45 p
ij -7- ее1ее 1 4- ееее2еnе е
jj2 n J ееееее N
j
T
k
1
ij
1 1
1 ееnее -3
jj-45 24+ ее2nее n p2 GJ2 - ееее1ееееее N 1 F1 J2 - ееее1ееееее ; еееее
е ; ееееее N J ееееее N
+
j
2n
2n 2 T T
k
1
1
5
1
5
2
3
1
3
1 1 1
1 ееnее - ее2ее
45 25+ ееnее n p2 GJ ееееее - ееееееееее N 1 F1 J ееееее - ееееееееее ; ееееее ; ееееее N J ееееее N
2
2n
2
2n 2 T T
2
5
1
5
1 3 1
1 ееnее - ее2ее
45 25+ ееnее n p2 GJ ееееее - ееееееееее N 1 F1 J ееееее - ееееееееее ; ееееее ; ееееее N J ееееее N
2
2n
2
2n 2 T T
1
2
1
1 1 1
1 ееnее -2
45 24+ ееnее n p2 GJ1 - ееееееееее N 1 F1 J1 - ееееееееее ; ееееее ; ееееее N J ееееее N
+
2n
2n 2 T T
1
2
1
1 3 1
1 ееnее -2
45 26+ ееnее n p2 GJ2 - ееееееееее N 1 F1 J2 - ееееееееее ; ееееее ; ееееее N J ееееее N
2n
2n 2 T T
3
1
3
1 3 1
1 ееnее - ее2ее zyzzyz
zzzz
n p GJ ееееее - ееееееееее N 1 F1 J ееееее - ееееееееее ; ееееее ; ееееее N J ееееее N
2
2n
2
2n 2 T T
{{
1
45 2
5+ ее2nее
2
And the classical SVC in reduced variables is
3
670
5.7 Second Virial Coefficient
bc = Coefficient@BStar, L, 0D
1
ееееееееееееееее4е
45 p
ij -7- ее1ее 1 4- ееее2еnе е
jj2 n J ееееее N
j
T
k
1
ij
n-3
n-3 1 1
1 ееnее - ее2ее
jj-45 27+ ее4nее p4 GJ ееееееее
еееееееее N 1 F1 J еееееееееееееееее ; ееееее ; ееееее N J ееееее N
+
j
2n
2n 2 T T
k
2
2
7
7
4
3 Hn - 1L
3 Hn - 1L 3 1
1 ееnее - ее2ее
45 28+ ееnее p4 GJ ееееееееееееееее
еееееееееее N 1 F1 J ееееееееееееееее
еееееееееее ; ееееее ; ееееее N J ееееее N
+
2n
2n
2 T T
4
3
45 27+ ееnее p4 GJ1 - ееееееееее N
2n
3 1 1
3 3 1
1 ееnее -3 yzzyzz
zzzz
JT 1 F1 J1 - ееееееееее ; ееееее ; ееееее N - 2 1 F1 J1 - ееееееееее ; ееееее ; ееееее NN J ееееее N
T
2n 2 T
2n 2 T
{{
2
The derived results are analytic expressions in terms of hypergeometric
functions 1 F1 allowing a graphical and analytical treatment of the SVC,
including quantum corrections. The representation of the second virial
coefficient up to second-order quantum corrections is thus given by
bstar = bc + /2 bq1 + /4 bq2;
To demonstrate the influence of the quantum mechanical corrections, let us
graphically examine the classical SVC, the two quantum corrections, and
the total representation of the SVC. We plot the reduced quantities
depending on the variable T * = T. Figure 5.7.23 shows the influence of the
first and second quantum correction on the SVC:
5. Quantum Mechanics
671
Plot@Evaluate@
8bc, bq1, bq2, bstar< Й. 8/ ▒ 1, V ▒ 1, n ▒ 6<D,
8T, 0.2, 10<, AxesLabel ▒ 8"T ", "B "<,
PlotStyle ▒ 8RGBColor@0, 0, 1D, RGBColor@0, 1, 0D,
RGBColor@1, 0, 0D, RGBColor@0, 0, 0D<, PlotRange ▒
82, 2<, Prolog ▒ 8Text@"Bc ", 81.6607, 0.650973<D,
Text@"Bq2 ", 80.660171, 0.452358<D,
Text@"Bq1 ", 80.861535, 0.704103<D,
Text@"B ", 82.08858, 0.916639<D<D;
B*
2
1.5
1
B*q1
0.5
*
2
-0.5 Bq2 B*
c *
B
-1
-1.5
-2
Figure 5.7.23.
4
6
8
10
T*
The figure contains the classical SVC (blue), the first quantum mechanical correction (red),
the second quantum correction of SVC (green), and the sum of the three parts (black). We
note that the second quantum corrections contains terms linear in s. Therefore, in addition
to L and n we have to specify the value of s.
For practical applications, it is sometimes necessary to have the numerical
values of the SVC and its first and second temperature derivatives
available. The numerical values of these quantities are tabulated in the
book by Hirschfelder et al. for the (12-6)-LJ potential. The first and
second derivative of B*c with respect to T * then follows by
672
5.7 Second Virial Coefficient
≥bc
b1 = T дддддддддддддд
≥T
jij 1
T jjjj ееееееееееееееее4е
j 45 p
k
4- ееее1ее е
jij
jj2-7- ее1nее J ееее1ее N 2 n
jj
T
j
k
jij
jj45 27+ ее4nее p4 GJ1 - ееее3ееееее N
jj
2n
j
k
3
3
ij
2 H1 - ееее
ееее L F H2 - ееее
ееее ; ее3ее ; ееее1е L
2n 1 1
2n 2 T
jj1 F1 J1 - ееее3ееееее ; ее1ееее ; ееее1ее N - ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееее +
j
T
2n 2 T
k
2
3
3
ее е L F H2 - ееее
ее е ; ееее5 ; ееее1е L y 1 ееnее -3
4 H1 - ееее
2n 1 1
2n 2 T z
+
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееее zzz J ееееее N
3 T2
{ T
2
5
4
2
7
n-3
n - 3 1 1 1 ееnее - ее2ее
45 27+ ееnее J ееееее - ееееее N p4 GJ ееееееееееееее ее N 1 F1 J ееееееееееееее ее ; еееее ; ееееее N J ееееее N
n
2
2n
2n 2 T T
4
2
7
3 Hn - 1L
3 Hn - 1L 3 1
еееееееееее N 1 F1 J ееееееееееееееее
еееееееееее ; ееееее ; ееееее N
45 28+ ееnее J ееееее - ееееее N p4 GJ ееееееееееееееее
n
2
2n
2n
2 T
2
5
4
2
3
1 ееnее - ее2ее
- 45 27+ ееnее J ееееее - 3N p4 GJ1 - ееееееееее N
J ееееее N
n
T
2n
2
3 1 1
3 3 1
1 ееnее -2
+
JT 1 F1 J1 - ееееееееее ; ееееее ; ееееее N - 2 1 F1 J1 - ееееееееее ; ееееее ; ееееее NN J ееееее N
T
2n 2 T
2n 2 T
ее2ее - ее3ее
4
n-3
n-3
45 27+ ееnее Hn - 3L p4 GH ееее
ееееее L F H ееее
ееееее + 1; ееее32 ; ееееT1е L H ееееT1е L n 2
2n 1 1 2n
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееее n
1
еееее
n
4
3 Hn - 1L
jij
jj45 28+ ееnее Hn - 1L p4 GJ ееееееееееееееее
ееееееееее N
2n
k
2 3 yy
3 Hn - 1L
5 1 1 ееnее - ее2ее yzzzzzzzz
zzzzzzzz + 1; еееее ; ееееее N J ееееее N
2n
2 T T
zz
{{{
еееееееееее
1 F1 J ееееееееееееееее
1
ееееееееееееееее4е
45 p
5- ееее
ij -7- ее1ее
2еnе е
jj2 n J4 - ееее1ееееее N J ееее1ее N
j
2n T
k
1
ij
n-3
jj-45 27+ ее4nее p4 GJ ееееееее
еееееееее N
j
2n
k
2
7
4
n - 3 1 1 1 ееnее - ее2ее
3 Hn - 1L
+ 45 28+ ееnее p4 GJ ееееееееееееееее
еееееееееее N
1 F1 J еееееееееееееееее ; ееееее ; ееееее N J ееееее N
2n 2 T T
2n
2
7
4
3 Hn - 1L 3 1 1 ееnее - ее2ее
3
еееееееееее ; еееее ; ееееее N J ееееее N
+ 45 27+ ееnее p4 GJ1 - ееееееееее N
1 F1 J ееееееееееееееее
2n
2 T T
2n
2
y
3 1 1
3 3 1
1 ееnее -3 yzzyzzzzz
zzzzzzz
JT 1 F1 J1 - ееееееееее ; ееееее ; ееееее N - 2 1 F1 J1 - ееееееееее ; ееееее ; ееееее NN J ееееее N
T
2n 2 T
2n 2 T
z
{{{
5. Quantum Mechanics
673
≥2 bc
дддддддд
b2 = T 2 дддддддддддддддд
≥T ≥T
ij
j 1
T 2 jjjj ееееееееееееееее4е
j 45 p
k
jij -7- ее1nее
jj2
k
3
3
ее е L F H2 - ееее
ее е ; ееее5 ; ееее1е L
4
3 i 8 H1 - ееее
jij
2n 1 1
2n 2 T
jj45 27+ ееnее p4 GJ1 - ееееееееее N jjjj- ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееее +
2n k
3 T3
k
3
3
3
ее е L H2 - ееее
ееее L F H3 - ееее
ееее ; ееее5 ; ее1ее L
4 H1 - ееее
2n
2n 1 1
2n 2 T
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееее 3 T3
3
3
3
8 H1 - ееее
ее е L H2 - ееее
ееее L F H3 - ееее
ееее ; ееее7 ; ее1ее L y
2n
2n 1 1
2n 2 T z
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееее zzz
15 T 4
{
2
4
2
3
1 ееnее -3
- 90 27+ ееnее J ееееее - 3N p4 GJ1 - ееееееееее N
J ееееее N
n
T
2n
3
3
ij
2 H1 - ееее
ееее L F H2 - ееее
ееее ; ее3ее ; ееее1е L
2n 1 1
2n 2 T
jj1 F1 J1 - ееее3ееееее ; ее1ееее ; ееее1ее N - ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееее +
j
T
2n 2 T
k
2
3
3
ее е L F H2 - ееее
ее е ; ееее5 ; ееее1е L y 1 ееnее -2
4 H1 - ееее
2n 1 1
2n 2 T z
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееее zzz J ееееее N
3 T2
{ T
4
2
7 2
5
n-3
n-3 1 1
45 27+ ееnее J ееееее - ееееее N J ееееее - ееееее N p4 GJ ееееееееееееее ее N 1 F1 J ееееееееееееее ее ; еееее ; ееееее N
n
2 n
2
2n
2n 2 T
2
3
4
2
7 2
5
1 ееnее - ее2ее
+ 45 28+ ееnее J ееееее - еееее N J ееееее - еееее N p4
J ееееее N
n
T
2 n
2
3 Hn - 1L
3 Hn - 1L 3
GJ ееееееееееееееее
еееееееееее N 1 F1 J ееееееееееееееее
еееееееееее ; ееееее ;
2n
2n
2
4
2
2
45 27+ ееnее J ееееее - 3N J еееее - 2N p4 GJ1 n
n
3
JT 1 F1 J1 - ееееееееее ;
2n
4
1 ji
2
еееее jjj45 27+ ееnее J еееее n
n
k
2
2
1 1
3 3 1
1 ееnее -1
ееееее ; ееееее N - 2 1 F1 J1 - ееееееееее ; ееееее ; ееееее NN J ееееее N
T
2 T
2n 2 T
7
n-3
ееееее N Hn - 3L p4 GJ еееееееееееееееее N
2
2n
n-3
3 1 1 ееnее - ее2ее yzz
zz + 1; ееееее ; ееееее N J ееееее N
2n
2 T T
{
2
1
1 F1 J ееееееееееееее ее
1
еееее
n
3
1
1 ееnее - ее2ее
ееееее N J ееееее N
+
T T
3
ееееееееее N
2n
ij
3
n-3
jj45 27+ ее4nее J ееее2е - еееее
е N Hn - 3L p4 GJ еееееееееееееееее N
j
n
2
2n
k
n-3
3 1 1 ееnее - ее2ее zyz
zz +
1 F1 J ееееееееееееее ее + 1; ееееее ; ееееее N J ееееее N
2n
2 T T
{
2
1
674
5.7 Second Virial Coefficient
1
еееее
n
ij
7
3 Hn - 1L
jj45 28+ ее4nее J ееее2е - еееее
е N Hn - 1L p4 GJ ееееееееееееееее
ееееееееее N
j
n
2
2n
k
3 Hn - 1L
5 1 1 ееnее - ее2ее yzz
zz +
еееееееееее + 1; еееее ; ееееее N J ееееее N
1 F1 J ееееееееееееееее
2n
2 T T
{
2
1
еееее
n
1
4
2
3
3 Hn - 1L
jij
jj45 28+ ееnее J еееее - ееееее N Hn - 1L p4 GJ ееееееееееееееее
ееееееееее N
n
2
2n
k
3 Hn - 1L
5 1 1 ееnее - ее2ее yzz
zz + 1; еееее ; ееееее N J ееееее N
2n
2 T T
{
2
1
еееееееееее
1 F1 J ееееееееееееееее
1
еееее
n
ij
n-3
n-3
jj15 28+ ее4nее J ееееееее
еееееееее + 1N Hn - 3L p4 GJ еееееееееееееееее N
j
2n
2n
k
n-3
5 1 1 ее2ее + ееnее zyz
zz +
1 F1 J ееееееееееееее ее + 2; ееееее ; ееееее N J ееееее N
2n
2 T T
{
1
1
еееее
n
2
ij
3 Hn - 1L
3 Hn - 1L
jj9 29+ ее4nее J ееееееееееееееее
еееееееееее + 1N Hn - 1L p4 GJ ееееееееееееееее
еееееееееее N
j
2n
2n
k
3 Hn - 1L
7 1 1 ее2ее + ееnее yzzyzz 1 4- ееее2еnе ее yzz
zzzz J ееееее N
zz еееееееееее + 2; еееее ; ееееее N J ееееее N
1 F1 J ееееееееееееееее
T
2n
2 T T
{{
{
1
1
ееееееееееееееее4е
45 p
2
1
jij
ji
jj2 2-7- ее1nее J4 - ееее1ееееее N jjj45 27+ ее4nее p4 GJ1 - ееее3ееееее N
jj
j
2 n jj
2n
j
k
k
3
3
2 H1 - ееее
ееее L F H2 - ееее
ееее ; ее3ее ; ееее1е L
ij
2n 1 1
2n 2 T
jj1 F1 J1 - ееее3ееееее ; ее1ееее ; ееее1ее N - ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееее +
j
T
2n 2 T
k
2
3
3
4 H1 - ееее
ее е L F H2 - ееее
ее е ; ееее5 ; ееее1е L y 1 ееnее -3
2n 1 1
2n 2 T z
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееее zzz J ееееее N
+
3 T2
{ T
2
5
4
2
7
n-3
n - 3 1 1 1 ееnее - ее2ее
45 27+ ееnее J ееееее - ееееее N p4 GJ ееееееееееееее ее N 1 F1 J ееееееееееееее ее ; еееее ; ееееее N J ееееее N
n
2
2n
2n 2 T T
4
2
7
3 Hn - 1L
3 Hn - 1L 3 1
еееееееееее N 1 F1 J ееееееееееееееее
еееееееееее ; ееееее ; ееееее N
45 28+ ееnее J ееееее - ееееее N p4 GJ ееееееееееееееее
n
2
2n
2n
2 T
2
5
4
2
3
1 ееnее - ее2ее
- 45 27+ ееnее J ееееее - 3N p4 GJ1 - ееееееееее N
J ееееее N
n
T
2n
2
3 1 1
3 3 1
1 ееnее -2
JT 1 F1 J1 - ееееееееее ; ееееее ; ееееее N - 2 1 F1 J1 - ееееееееее ; ееееее ; ееееее NN J ееееее N
+
T
2n 2 T
2n 2 T
4
ее2ее - ее3ее
n-3
n-3
45 27+ ееnее Hn - 3L p4 GH ееее
ееееее L F H ееее
ееееее + 1; ееее32 ; ееееT1е L H ееееT1е L n 2
2n 1 1 2n
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееее n
5. Quantum Mechanics
1
еееее
n
675
ij
3 Hn - 1L
jj45 28+ ее4nее Hn - 1L p4 GJ ееееееееееееееее
ееееееееее N
j
2n
k
2 3 y
3 Hn - 1L
5 1 1 ееnее - ее2ее yzzzzz
zzzzz
F
J
ееееееееееееееее
ееее
е
еее
е
ее
+
1;
ееее
е
;
ееее
е
е
N
J
ееее
е
е
N
1 1
2n
2 T T
z
{{
1 y
1
1 5- ееее2еnе е zzz
zz + еееееееееееееееее
J ееееее N
zz 45 p4
T
{
ij -7- ее1ее
jj2 n J4 - ееее1ееееее N J5 - ееее1ееееее N
j
2n
2n
k
ij
n-3
n-3 1 1
1 ееnее - ее2ее
jj-45 27+ ее4nее p4 GJ ееееееее
еееееееее N 1 F1 J еееееееееееееееее ; ееееее ; ееееее N J ееееее N
+
j
2n
2n 2 T T
k
2
7
2
7
4
3 Hn - 1L
3 Hn - 1L 3 1
1 ееnее - ее2ее
45 28+ ееnее p4 GJ ееееееееееееееее
еееееееееее N 1 F1 J ееееееееееееееее
еееееееееее ; ееееее ; ееееее N J ееееее N
+
2n
2n
2 T T
4
3
3 1 1
45 27+ ееnее p4 GJ1 - ееееееееее N JT 1 F1 J1 - ееееееееее ; еееее ; ееееее N 2n
2n 2 T
2
1 y
3 3 1
1 ееnее -3 yzz 1 6- ееее2еnе е yzzzzz
zz J ееееее N
zzzzz
2 1 F1 J1 - ееееееееее ; ееееее ; ееееее NN J ееееее N
T
T
2n 2 T
z
{
{{
The first few lines of table I-B contained in the appendix of Hirschfelder et
al. then follows by
676
5.7 Second Virial Coefficient
t1 = Table@N@8T, bc, b1, b2, b1 - bc< Й. n ф 6, 9D, 8T, .3, 1, .05<D;
PrependTo@t1, 8"T", "B*c ", "b1", "b2", "b1-B*c "<D;
TableForm@Map@Map@PaddedForm@#, 85, 2<D &, #D &, t1DD
T
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
B*c
-27.88
-18.75
-13.80
-10.75
-8.72
-7.27
-6.20
-5.37
-4.71
-4.18
-3.73
-3.36
-3.05
-2.77
-2.54
Comparing the calculated
excellent agreement. The
above are not restricted to
the exponent n > 3. For
(16-8)-potential by
b1
76.61
45.25
30.27
21.99
16.92
13.58
11.25
9.55
8.26
7.25
6.45
5.80
5.26
4.81
4.43
b2
-356.88
-189.47
-116.37
-78.88
-57.34
-43.88
-34.92
-28.64
-24.06
-20.61
-17.94
-15.83
-14.12
-12.71
-11.54
b1-B*c
104.49
64.00
44.07
32.74
25.64
20.86
17.45
14.91
12.97
11.43
10.19
9.17
8.31
7.59
6.97
figures with Hirschfelder's result demonstrates
analytical results derived in the calculations
the (12-6)-LJ potential but allow any choice of
example, we can determine the SVC for a
5. Quantum Mechanics
677
t2 = Table@N@8T, bc, b1, b2, b1 - bc< Й. n ф 8, 9D, 8T, 1, 2, .05<D;
PrependTo@t2,
8"T", "\!\H\HB\_c\%*\L\L", "b1", "b2", "b1-\!\H\HB\_c\%*\L\L"<D;
TableForm@HHPaddedForm@#1, 85, 4<D &L ЙШ #1 &L ЙШ t2D
T
1.0000
1.0500
1.1000
1.1500
1.2000
1.2500
1.3000
1.3500
1.4000
1.4500
1.5000
1.5500
1.6000
1.6500
1.7000
1.7500
1.8000
1.8500
1.9000
1.9500
2.0000
B*c
-1.0453
-0.9233
-0.8157
-0.7201
-0.6348
-0.5580
-0.4887
-0.4259
-0.3686
-0.3162
-0.2681
-0.2238
-0.1828
-0.1449
-0.1097
-0.0769
-0.0463
-0.0177
0.0091
0.0342
0.0579
b1
2.6042
2.4020
2.2270
2.0742
1.9397
1.8205
1.7142
1.6188
1.5328
1.4549
1.3839
1.3191
1.2597
1.2050
1.1545
1.1077
1.0643
1.0239
0.9862
0.9509
0.9179
b2
-6.9604
-6.3427
-5.8170
-5.3651
-4.9732
-4.6305
-4.3288
-4.0612
-3.8226
-3.6087
-3.4160
-3.2415
-3.0828
-2.9380
-2.8055
-2.6836
-2.5713
-2.4674
-2.3711
-2.2817
-2.1983
We also can represent the data graphically:
b1-B*c
3.6495
3.3253
3.0427
2.7943
2.5745
2.3786
2.2029
2.0447
1.9014
1.7710
1.6520
1.5429
1.4425
1.3499
1.2642
1.1847
1.1107
1.0416
0.9771
0.9167
0.8600
678
5.7 Second Virial Coefficient
Plot@Evaluate@8bc, b1, b2, b1 - bc< Й. n ф 8D,
8T, .3, 2<, AxesLabel ф 8"T * ", "B* "<, PlotStyle ф
8RGBColor@0.996109, 0, 0D, RGBColor@0.500008, 0.996109, 0D,
RGBColor@0.500008, 0, 0.250004D, RGBColor@0, 0, 0.996109D<,
Prolog ф 8Text@"B*c ", 80.224533, -12.4014<D, Text@"B*1 ",
80.224533, 34.4014<D, Text@"B*2 ", 80.425166, -30.6213<D,
Text@"B*1 -B*c ", 80.564147, 20.2026<D<D;
B*
30
B*1
B*1 -B*c
20
10
-10
B*c
0.5
1
1.5
2
T*
-20
-30
B*2
Knowing the analytical expressions of the SVC, we are able to calculate
either numerical values of of the classical SVC and its derivatives or
represent the data graphically. We are not only restricted to classical
values but can incorporate the quantum mechanical corrections. The first
and second temperature derivatives for B*q are
bqq1 = T ≥T bstar;
bqq2 = T2 ≥T,T bstar;
A table containing the SVC with quantum corrections and the two
derivatives is generated by
5. Quantum Mechanics
679
t3 = Table@
N@8T, bstar, bqq1, bqq2, bqq1 - bstar< Й. 8n ф 8, s ф 1, L ф 1<, 9D,
8T, 1, 2, .05<D; PrependTo@t3, 8"T", "B* ", "B*1 ", "B*2 ", "B*1 -B* "<D;
TableForm@Map@Map@PaddedForm@#, 85, 4<D &, #D &, t3DD
T
1.0000
1.0500
1.1000
1.1500
1.2000
1.2500
1.3000
1.3500
1.4000
1.4500
1.5000
1.5500
1.6000
1.6500
1.7000
1.7500
1.8000
1.8500
1.9000
1.9500
2.0000
B*
-1.3947
-1.2481
-1.1181
-1.0022
-0.8985
-0.8051
-0.7208
-0.6443
-0.5746
-0.5108
-0.4524
-0.3987
-0.3491
-0.3032
-0.2606
-0.2210
-0.1842
-0.1497
-0.1175
-0.0873
-0.0590
B*1
3.1190
2.8954
2.6961
2.5181
2.3587
2.2156
2.0866
1.9700
1.8643
1.7680
1.6801
1.5995
1.5256
1.4574
1.3945
1.3362
1.2821
1.2317
1.1848
1.1409
1.0998
B*2
-7.8531
-7.3252
-6.8361
-6.3879
-5.9797
-5.6087
-5.2718
-4.9658
-4.6874
-4.4338
-4.2022
-3.9903
-3.7959
-3.6173
-3.4527
-3.3007
-3.1600
-3.0296
-2.9084
-2.7955
-2.6902
These values are graphically represented by
B*1 -B*
4.5137
4.1435
3.8141
3.5203
3.2572
3.0207
2.8074
2.6143
2.4388
2.2788
2.1325
1.9982
1.8746
1.7606
1.6551
1.5572
1.4662
1.3814
1.3023
1.2282
1.1588
680
5.7 Second Virial Coefficient
Plot@Evaluate@8bstar, bqq1, bqq2, bqq1 - bc< Й. 8n ф 8, s ф 1, L ф 1<D,
8T, .3, 2<, AxesLabel ф 8"T * ", "B* "<, PlotStyle ф
8RGBColor@0.996109, 0, 0D, RGBColor@0.500008, 0.996109, 0D,
RGBColor@0.500008, 0, 0.250004D, RGBColor@0, 0, 0.996109D<,
Prolog ф 8Text@"B* ", 80.304995, 40.1747<D, Text@"B*1 ",
80.466965, -26.9214<D, Text@"B*2 ", 80.578254, 32.1998<D,
Text@"B*1 -B*c ", 80.901671, 12.5162<D<,
PlotLabel ф "s=1, L=1"D;
B*
30
s=1, L=1
B*2
20
10
-10
-20
-30
B*1 -B*c
0.5
1
1.5
2
T*
B*1
The first and second derivatives of B* with respect to T * are of practical
importance.
5.7.4 Shape Dependence of the Boyle Temperature
Stogryn and Hirschfelder [5.16] showed that the SVC can be separated
into a bound state, a meta-stable state, and a continuum state contribution.
For the (12-6)-LJ potential, they gave the temperature dependence of these
contributions in tabular form.
At low temperatures, the average energies of the colliding molecules are of
the order of the energy of the well depth. The molecule spends much time
in the bound region of the molecular potential. Mutual attraction of the
molecules results in a decrease of pressure, and the SVC is negative.
5. Quantum Mechanics
681
At high temperatures, corresponding to high energies compared to the well
depth, the main contribution comes from the repulsive branch of the
potential. Repulsion increases the pressure and SVS becomes negative.
From the above-mentioned investigation of the SVC for the (12-6)-LJ
potential follows that the SVC for the bound states and the meta-stable
states remains positive, whereas the contribution by the continuum states
becomes negative and equals the bound state and metastable state
contribution at the Boyle temperature leading to BHTL = 0. The shape
dependence of the SVC on the exponent n is shown in Figure 5.7.24.
ns = 84.0, 4.5, 5, 5.5, 6, 6.5, 7, 7.5<;
Plot@Evaluate@Map@bc Й. n ▒ # &, nsDD,
8T, 1, 300<, AxesLabel ▒ 8"T ", "Bq0 "<,
PlotStyle ▒ RGBColor@0, 0, 0.996109D,
TextStyle ▒ 8FontFamily ▒ "Arial",
FontSize ▒ 15, FontWeight ▒ "Bold"<,
AxesStyle ▒ 8Thickness@0.005D<,
Prolog ▒ 8Text@"n=4.0", 8161.363, 0.171682<D,
Text@"n=7.5", 8161.363, 0.6<D<D;
B*q0
0.6
n=7.5
0.4
0.2
-0.2
Figure 5.7.24.
n=4.0
T*
50 100 150 200 250 300
The scaled SVC for different potential orders n.
The Boyle temperatures are calculated by solving the defining equation
BHTB L = 0. The solution is carried out by the function FindRoot[]:
682
5.7 Second Virial Coefficient
BoyleT = HFindRoot@#1 == 0, 8T, 10<D &L ЙШ
Hbc Й. n ф #1 &L ЙШ Table@i, 8i, 3.1, 7.5, .1<D;
The following table collects the Boyle-temperatures for different values of
n:
tabBoyle = 8Table@i, 8i, 3.1, 7.5, .1<D, T Й. BoyleT<T ;
tb = Prepend@tabBoyle, 8"n", "TB "<D;
A graphical representation of these numerical values is given in the
following plot:
ListPlot@tabBoyle, AxesLabel ▒ 8"n", "TB "<,
PlotStyle ▒ RGBColor@0.996109, 0, 0DD;
TB
50
40
30
20
10
4
5
6
7
n
The result is that the Boyle temperaure is a decreasing function which has
a singularity at n = 3. For n values much larger than 3, the Boyle
temperature approaches zero.
At often unphysically high temperatures, the molecules collide with such
high energies that they interpenetrate each other. They behave as if they
had a smaller volume so that B(T) goes through a maximum. This is shown
in Figure 5.7.24. To determine the change of this maximum by altering the
potential order, we determine the coresponding temperature values by
5. Quantum Mechanics
683
Tmax = HFindRoot@#1 == 0, 8T, 20<D &L Й@
H≥T bc Й. n ▒ #1 &L Й@ Table@i, 8i, 3.2, 7.5, .1<D;
FindRoot::lstol :
The line search decreased the step size to within tolerance specified by AccuracyGoal
and PrecisionGoal but was unable to find a sufficient decrease in
the merit function. You may need more than MachinePrecision
digits of working precision to meet these tolerances. More?
FindRoot::lstol :
The line search decreased the step size to within tolerance specified by AccuracyGoal
and PrecisionGoal but was unable to find a sufficient decrease in
the merit function. You may need more than MachinePrecision
digits of working precision to meet these tolerances. More?
These maximum temperatures are collected in the following table. For He,
this maximum was observed experimentally near 200 K.
tabTmax =
Transpose@8Table@i, 8i, 3.2, 7.5, .1<D, T Й. Tmax<D;
tT = Prepend@tabTmax, 8"n", "TB "<D;
The graphical representation of these data is given in the following plot:
ListPlot@tabTmax, AxesLabel ▒ 8"n", "Tmax "<,
PlotStyle ▒ RGBColor@0.996109, 0, 0D,
PlotRange ▒ 883, 7.7<, 80, 220<<D;
Tmax
200
150
100
50
4
5
6
7
n
684
5.7 Second Virial Coefficient
5.7.5 The High-Temperature Partition Function for Diatomic
Molecules
The partition function of a diatomic molecule is important for many
applications from astrophysics to reaction kinetics. In courses on physical
chemistry, it is treated in the harmonic oscillator approximation ? rigid
rotator approximation, and anharmonicity and rotation ? vibration
interactions are included in the spirit of the JANAF tables.
It is known from high-temperature chemistry that for accurate
thermodynamic functions, bound states from the solution of the
rotation?vibration SchrЖdinger equation of the molecule
≥ ynJ
J HJ +1L я
ееееееее
≥r2еееее - JUHrL + ееееееееееееееее
2 m r2еееее - EnJ N ynJ = 0
2
2
(5.7.119)
where ynJ are rotation?vibration eigenfunctions, n and J are vibrational
and rotational quantum numbers, respectively, m represents the reduced
mass, and EnJ , the rotation?vibration eigenvalues, must be calculated. The
meta-stable states behind the rotational barrier must be included.
Mies and Julienne [5.18] investigated the statistical thermodynamic of the
diatomic molecule using numerical techniques for the exact scattering
theory of the SVC.
For the equilibrium reaction
X2 V 2 X
as an example, they showed that the concentration equilibrium constant Kc
can be expressed by the SVC
BHTL = - Kc
(5.7.120)
As for real molecules and atoms excited, often degenerate electronic states
must be included, they defined a generalized SVC by
XB\ = H?i Bi HtL gi Hx2 L ?-Eij ЙHkB TL L H? j g j HxL ?-Eij ЙHkB TL L,
where Bi is the SVC for a molecular state i, g is the electronic degeneracy
Eij is the excitation energy, g j is the electronic degeneracy of the atomic
5. Quantum Mechanics
685
state j and E j is its excitation energy. Phair, Birlsi and Holland [5.17]
derived the partition function from
K
1
qHX2 L
ееееееее
еееее
V
K p = ееее
еcееее = ееее
ееееее ееееееее
ееееееее ,
qHX L 2
RT
kB T I ееееееее
V ееее M
with K p the pressure equilibrium constant, V the volume of the system,
qHX L the monomer partition function, and qHX2 L the dimer parition
function.
As qHX L depends only on mass, temperature, volume, and electronic
degeneracy g, the diatomic partition function for the bound state can be
written
2pm k T 3 V
x B
-D0 ЙHkB TL
2
,
qHX2 L = -BHTL I ееееееееееееееее
h2 ееееееее M ееее
NеAеее g0 HX L ?
(5.7.121)
where D0 is the spectroscopic dissociation energy of the ground electronic
state of the molecule X2 , if the energy zero is taken as the lowest
vibrational state (one can take as well De as energy zero). If we insert the
analytical results for BHTL for the H2 n - nL-potential derived above a
closed-form representation of a realistic partition function including
rotation?vibration coupling, anharmonicity up the disociation limit,
meta-stable states behind the rotational barrier, and the continuum or
scattering states.
From the diatomic partition function, the molecular thermodynamic
functions can be calculated by standard methods. Phais et al. [5.17] gave
explicite formulas for
B*
1
ее + ?M
HT0 - H00 = R T I4 + ееее
B*
*
*
B
B
B*1 2 yz
i
1
2
е
е
+
ееее
е
е
I
ееее
ее M .
C0p = R j4 + 2 ееее
*
*
B
B
B* {
k
(5.7.122)
(5.7.123)
Equation (5.7.122) scaled by R reads
2
2 b1
b2
i b1 y
Cp = - jj ддддддддд zz + дддддддддддддд + ддддддддд + 4;
bc
bc
k bc {
A graphical representation of this function for differet values of n is given
next.
686
5.7 Second Virial Coefficient
pl1 =
Plot@Evaluate@Map@Cp Й. n ▒ # &, 83.5, 4, 5, 6, 7<DD,
8T, .01, .3<, AxesLabel ▒ 8"T ", "Cp ЙR"<,
PlotStyle ▒ RGBColor@0, 0, 0.996109D,
Prolog ▒ 8Text@"n=3.5", 80.208115, 6.37933<D,
Text@"n=7", 80.120138, 4.74692<D<D;
Cp ЙR
6.5
n=3.5
6
5.5
5
n=7
4.5
0.05
0.1
0.15
0.2
0.25
0.3
T*
Phair et al. used a five-parameter Hulburt?Hirschfelder potential in their
numerical calculations for Bq0 . The following set of data is taken from
their article representing the scaled C p values [5.17].
data = 880.0174, 4.5<, 80.0384, 4.57<, 80.0522, 4.69<,
80.0696, 4.75<, 80.0869, 5.035<, 80.1043, 5.52<,
80.1217, 5.99<, 80.1304, 6.14<, 80.139, 6.20<,
80.147, 6.19<, 80.156, 6.09<, 80.174, 5.75<,
80.191, 5.28<, 80.208, 4.83<, 80.217, 4.62<<;
A combination of our symbolic calculations and their numerical results
demonstrates a qualitative agreement. The results are shown in the Figure
5.7.25.
5. Quantum Mechanics
687
Show@
8pl1, ListPlot@data, DisplayFunction > IdentityD<,
DisplayFunction > $DisplayFunctionD;
Cp ЙR
6.5
n=3.5
6
5.5
5
n=7
4.5
0.05
Figure 5.7.25.
0.1
0.15
0.2
0.25
0.3
T*
Shape dependence of the ``dissociation'' maximum of the heat capacity C p . The points
denoted by dots are for N2 teken from Phair et al, using the five parameter
Hulburt?Hirschfelder potential in the numerical calculation of Bq0 and its temperature
derivatives.
5.8 Exercises
1. Examine the spectrum of the eigenvalues for a potential well with
different depths. Give a graphical representation of the eigenvalues
depending on different depths.
2. Determine the wave functions for different eigenvalues for the
potential well by using the methods discussed in Section 5.3.
3. Check the relation ╩ a ╩2 + ╩ b ╩2 = 1 for the anharmonic oscillator.
4. Reexamine the PЖschel?Teller problem and study the expectation
values Xxn \ given by
Xxn \ = ? y* xn y dx
for different values of n.
5. Plot the radial part of the wave function of the hydrogen atom for
different quantum numbers n and l. Examine the influence of the
charge Z.
688
5.8 Exercises
6. Create a graphical representation of the f orbital for the europium
atom.
5.9 Packages and Programs
5.9.1 Package QuantumWell
This package serves to examine a one-dimensional quantum dot.
BeginPackage@"QuantumWell`"D;
Clear@PsiSym, PsiASym, SpectrumD;
PsiSym::usage =
"PsiSym@x_,k_,a_D determines the symmetric
eigenfunction for a potential well of depth V0. The input parameter k fixes the energy and
2a the width of the well. PsiSym is useful for
a numerical representation of eigenfunctions.";
PsiASym::usage =
"PsiASym@x_,k_,a_D determines the antisymmetric
eigenfunction for a potential well of depth V0. The input parameter k fixes the energy and 2
a the width of the well. PsiASym is useful for
a numerical representation of eigenfunctions.";
Spectrum::usage =
"Spectrum@V0_,a_D calculates the negative
eigenvalues in a potential well. V0 is the
potential depth and 2a the width of the
well. The eigenvalues are returend as a list
and are available in the variables lsym and
lasym as replacement rules. The corresponding
plots of eigenfunctions are stored in the
variables Plsym and Plasym. The determining
equation for the eigenvalues is plotted.";
5. Quantum Mechanics
Hdefine global variablesL
Plsym::usage = "Variables containing the
symmetric plots of the eigenfunctions.";
Plasym::usage = "Variables containing the
antisymmetric plots of the eigenfunctions.";
lsym::usage = "List of symmetric eigenvalues.";
lasym::usage = "List of antisymmetric eigenvalues.";
k : usage = "Eigenvalue.";
Begin@"`Private`"D;
Hsymmetric eigenfunctionsL
PsiSym@x_, k_, a_D := Block@8<,
Hdefine the three domains of solutionL
Which@Infinity < x && x < a,
1 Й Sqrt@a Exp@2 a k Tan@k aDD H1 + 1 Й Hk Tan@k aD aL +
k Tan@k aD Й Hk ^ 2 aL + Hk Tan@k aDL ^ 2 Й k ^ 2LD
Exp@k Tan@k aD xD, a ├ x && x ├ a,
1 Й Sqrt@a Exp@2 a k Tan@k aDD H1 + 1 Й Hk Tan@k aD aL +
k Tan@k aD Й Hk ^ 2 aL + Hk Tan@k aDL ^ 2 Й k ^ 2LD
Exp@k Tan@k aD aD Cos@k xD Й Cos@k aD,
a < x && x < Infinity, 1 Й Sqrt@a Exp@2 a k Tan@k aDD
H1 + 1 Й Hk Tan@k aD aL + k Tan@k aD Й Hk ^ 2 aL +
Hk Tan@k aDL ^ 2 Й k ^ 2LD Exp@k Tan@k aD xDDD;
Hantisymmetric eigenfunctionsL
PsiASym@x_, k_, a_D := Block@8<,
Hdefine the three domains of solutionL
Which@Infinity < x && x < a,
1 Й Sqrt@a Exp@2 a Hk Cot@k aDLD
H1 + 1 Й Hk Cot@k aD aL + Hk Cot@k aDL Й Hk ^ 2 aL +
Hk Cot@k aDL ^ 2 Й k ^ 2LD Exp@Hk Cot@k aDL xD,
a ├ x && x ├ a, 1 Й Sqrt@a Exp@2 a Hk Cot@k aDLD
H1 + 1 Й Hk Cot@k aD aL + Hk Cot@k aDL Й Hk ^ 2 aL +
Hk Cot@k aDL ^ 2 Й k ^ 2LD
Exp@Hk Cot@k aDL aD Sin@k xD Й Sin@k aD,
a < x && x < Infinity,
1 Й Sqrt@a Exp@2 a Hk Cot@k aDLD
H1 + 1 Й Hk Cot@k aD aL + Hk Cot@k aDL Й Hk ^ 2 aL +
Hk Cot@k aDL ^ 2 Й k ^ 2LD
689
690
5.9 Packages and Programs
Exp@Hk Cot@k aDL xDDD;
Hdetermination of the eigenvalues;
plot of the eigenfunctionsL
Spectrum@V0_, a_D :=
Block@8hbar = 1, m = 1, ymax, C2, rhs, lhssym,
lhsasym, equatsym, equatasym, kmax, nsym,
nasym, resultsym, resultasym<, Hdefine
constants and the eigenvalue equationL
C2 = 2 m V0 a ^ 2 Й Hhbar^ 2L;
rhs = Tan@k aD;
lhssym = Sqrt@C2 Hk aL ^ 2D Й Hk aL;
lhsasym = k a Й Sqrt@C2 Hk aL ^ 2D;
equatsym = Sqrt@C2 Hk aL ^ 2D Й Hk aL Tan@k aD;
equatasym = k a Й Sqrt@C2 Hk aL ^ 2D Tan@k aD;
Hlocation of the singularity in kL
kmax = Sqrt@C2 Й a ^ 2D;
Hnumber of symmetric eigenvaluesL
nsym = Floor@N@kmax Й HPi Й aLDD + 1;
Hnumber of antisymmetric eigenvaluesL
nasym = Floor@N@Hkmax Pi Й H2 aLL Й HPi Й aLDD + 1;
Hinitialize the
lists for the eigenvaluesL
lsym = 8<;
lasym = 8<;
Hcalculate the symetric eigenvaluesL
Do@resultsym = Chop@FindRoot@
equatsym m 0, 8k, 0.1 + HPi Й aL Hi 1L<DD;
AppendTo@lsym, resultsymD, 8i, 1, nsym<D;
HChop@D replaces
small numbers H<10^H10LL by 0L
Hcalculate the antisymmetric eigenvaluesL
Do@resultasym = Chop@FindRoot@equatasym m 0,
8k, Pi Й H2 aL + 0.1 + HPi Й aL Hi 1L<DD;
AppendTo@lasym, resultasymD, 8i, 1, nasym<D;
Hplot the eigenvalue equationL
ymax = lhssym 1.5 Й. lsym@@1DD;
Off@Plot::plnrD;
Plot@8rhs, lhssym, lhsasym<,
8k, 0.01, 3 kmax Й 2<, PlotRange ▒ 8ymax, ymax<,
Prolog ▒ Thickness@0.001D,
PlotStyle ▒ 8RGBColor@1, 0, 0D, Dashing@8<D,
5. Quantum Mechanics
Dashing@81 Й 60<D<, AxesLabel ▒ 8"k", " "<D;
On@Plot::plnrD;
Hplot the symmetric eigenfunctionsL
Do@k1 = k Й. lsym@@iDD;
Plsym@iD = Plot@PsiSym@x, k1, aD, 8x, 2 a, 2 a<,
AxesLabel ▒ 8"x", "\!\H\\^s\L\n"<, PlotLabel >
"
\!\Hk\_i\L= " <> ToString@k1D,
Frame > True, PlotRange ▒ All,
Prolog ▒ Thickness@0.001D, PlotStyle ▒
8Dashing@81 Й Hi 20L<D<D, 8i, 1, nsym<D;
Hplot the antisymmetric eigenfunctionsL
Do@k1 = k Й. lasym@@iDD;
Plasym@iD = Plot@PsiASym@x, k1, aD, 8x, 2 a, 2 a<,
AxesLabel ▒ 8"x", "\!\H\\^a\L\n"<,
PlotLabel > "
\!\Hk\_i\L= " <> ToString@k1D,
Frame > True, PlotRange ▒ All,
Prolog ▒ Thickness@0.001D, PlotStyle ▒
8Dashing@81 Й Hi 20L<D<D, 8i, 1, nasym<D;
Hprint the eigenvaluesL
Print@" "D;
Print@" eigenvalues "D;
Print@" "D;
Do@k1 = k Й. lsym@@iDD;
If@i ├ nasym, k2 = k Й. lasym@@iDD, k2 = ""D;
Print@" sym eigenvalue k",
i, " = ", k1, " asym eigenvalue k",
i, " = ", k2D, 8i, 1, nsym<DD;
End@D;
EndPackage@D;
Set::patset : Warning: k : usage in assignment
k : usage = Eigenvalue. represents a named pattern;
use symbol::tag to represent a message name. More?
Here are some tests of the symmetric and antisymmetric wave function.
691
692
5.9 Packages and Programs
ys
Plot@8PsiSym@x, 1.30183, 1D,
PsiSym@x, 3.818578969739773`, 1D<, 8x, 2., 2<,
Frame > True, FrameLabel > 8"x", "\s "<,
PlotStyle > 8RGBColor@1, 0, 0D, RGBColor@0, 0, 1D<,
Prolog > 88RGBColor@1, 0, 0D, Text@"k1 =1.3018",
81., 0.220252<D<, 8RGBColor@0, 0, 1D,
Text@"k2 =3.8185", 81., 0.420252<D<<D;
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
k1 =1.3018
k2 =3.8185
-2
-1
0
x
1
2
5. Quantum Mechanics
693
ya
Plot@8PsiASym@x, 2.5856031391976373`, 1D,
PsiASym@x, 4.851591489119471`, 1D<, 8x, 2., 2<,
Frame > True, FrameLabel > 8"x", "\a "<,
PlotStyle > 8RGBColor@1, 0, 0D, RGBColor@0, 0, 1D<,
Prolog > 88RGBColor@1, 0, 0D, Text@"k1 =2.5856",
81.2, 0.220252<D<, 8RGBColor@0, 0, 1D,
Text@"k2 =4.8515", 81.2, 0.420252<D<<D;
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
k1 =2.5856
k2 =4.8515
-2
-1
0
x
1
2
5.9.2 Package HarmonicOscillator
The package HarmonicOscillator provides functions to represent
eigenfunctions of the harmonic oscillator.
BeginPackage@"HarmonicOscillator`"D;
Clear@a, across, Psi, wcl, wqmD;
Psi::usage =
"Psi@xi_,n_D represents the eigenfunction
of the harmonic oscillator. The first
argument xi is the spatial coordinate. The
second argument n fixes the eigenstate.";
wcl::usage =
"wcl@xi_,n_D calculates the classical probability
694
5.9 Packages and Programs
of locating the particle in the harmonic
potential. The first argument xi is the
spatial coordinate while n determines
the energy given as eigenvalue.";
wqm::usage =
"wqm@xi_,n_D calculates the quantum mechanical
probability for an eigenvalue state n. The first
argument represents the spatial coordinate.";
a::usage = "a@psi_, xi_:xD annihilation operator for
eigenfunction psi. The second argument specifies
the independent variable of the function psi.";
across::usage =
"across@psi_, xi_:xD creation operator for
eigenfunction psi. The second argument
specifies the independent variable of psi.";
x::usage;
Begin@"`Private`"D;
Heigenfunctions of the harmonic oscillatorL
Psi@xi_, n_D :=
HermiteH@n, xiD Exp@xi ^ 2 Й 2D Й Sqrt@n ! 2 ^ n Sqrt@PiDD;
\!\H\H\ \_ n_@[_D := Psi@[, nD;\L\L
Hclassical probability distribution
of the harmonic oscillatorL
wcl@xi_, n_D := 1 Й HSqrt@2 n + 1D
Sqrt@1 Hxi Й Sqrt@2 n + 1DL ^ 2D 2 PiL;
Hquantummechanical probability
distribution of the harmonic oscillatorL
wqm@xi_, n_D := Psi@xi, nD ^ 2;
Hannihilation operatorL
a@psi_, xi_: xD := Hxi psi + D@psi, xiDL Й Sqrt@2D;
Hcreation operatorL
5. Quantum Mechanics
695
across@psi_, xi_: xD := Hxi psi D@psi, xiDL Й Sqrt@2D;
End@D;
EndPackage@D;
5.9.3 Package AnharmonicOscillator
The package AnharmonicOscillator serves to determine the properties of
the PЖschel?Teller problem.
BeginPackage@"AnharmonicOscillator`"D;
Clear@AsymptoticPT, PlotPT, PoeschelTellerD;
PoeschelTeller::usage =
"PoeschelTeller@x_, n_, indexN_D calculates the
eigenfunction of the Poeschel Teller potential
for discrete eigenvalues.N determines the
depth of the potential V0 Sech@xD by V0=NH
N+1L.n fixes the state where 0 < n <= N.";
w1a::usage = "The variable contains the
analytic expression for the asymptotic
approximation for x > Infinity.";
w2a::usage = "The variable contains the
analytic expression for the asymptotic
approximation for x > Infinity.";
Transmission::usage =
"Variable containing the expression
for the transmission coefficient. The
independent variables are N and k.";
Reflection::usage =
"Variable containing the reflection coefficient.
The independent variables are N and k.";
AsymptoticPT::usage =
"AsymptoticPT@indexN_,kin_D determines the
asymptotic approximation for ╩x╩>Infinity
for the continuous case of eigenvalues in
696
5.9 Packages and Programs
a Poeschel Teller potential. The function
yields an analytic expression for╩bHkL╩^2.
The variables Transmission and Reflection
contain the expressionsfor the transmission
and the reflection coefficients. w1a and
w2a contain the approximations for x>
Infinity and x>Infinity, respectively.";
PlotPT::usage =
"PlotPT@kini_,kend_,type_D gives a graphical
representation of the reflection or transmission
coefficient depending on the value of
thevariable type. If type is set to
the string r the reflection coefficient
isplotted. If type is set to the
transmission coefficient is represented.
This function creates 5 different curves.";
Begin@"`Private`"D;
Hdefine the eigenfunctionsL
PoeschelTeller@x_, n_Integer, indexN_IntegerD :=
Block@8norm, integrand, xi<,
If@n ├ indexN && n > 0, Heigenfunctions are
the associated Legendre polynomialsL
integrand = LegendreP@indexN, n, xiD;
Hcalculate the normalization constantL
norm =
Integrate@integrand^ 2 Й H1 xi ^ 2L, 8xi, 1, 1<D;
Hnormalize and simplify the functionsL
integrand = integrand Й Sqrt@normD Й. xi ▒ Tanh@xD;
Simplify@integrandD,
Herror conditionsL
If@indexN < n,
Print@" wrong argument! use n > N"DD;
If@n < 0, Print@" wrong
argument! use n < 0"DDDD;
Hasymptotic expansionL
AsymptoticPT@indexN_, kin_D :=
Block@8k, rule1, rule2, wavefkt1, wavefkt2,
asympt1, w1, asymt2, w2, akh, bkh, ak<,
5. Quantum Mechanics
Hreplacement rules for the parametersL
rule1 = 8a ▒ 1 Й 2 I k + H1 Й 4 + V0L ^ H1 Й 2L,
b ▒ 1 Й 2 I k H1 Й 4 + V0L ^ H1 Й 2L, c ▒ 1 I k<;
rule2 = 8V0 ▒ indexN H1 + indexNL<;
wavefkt1 = ak HH1 xi ^ 2L Й 4L ^ HI k Й 2L;
wavefkt2 = Hypergeometric2F1@a, b, c, H1 + xiL Й 2D;
Hasymptotic expansion for x▒Infinity,
equation 5.5 .63L
asymt1 = Series@wavefkt2, 8xi, 1, 0<D;
w1 = wavefkt1 Normal@asymt1D Й. rule1;
w1 = w1 Й. rule2;
w1 = w1 Й. xi ▒ Tanh@xD;
w1 = Simplify@w1D;
w1 = w1 Й. Sech@xD ▒ 2 Exp@xD;
w1a = PowerExpand@w1D;
Hasymptotic expansion for x▒
Infinity by equation 5.5 .655.5 .68L
asymt2 = Series@wavefkt2, 8xi, 1, 1<D;
Hinvert substitutionL
w2 = wavefkt1 Normal@asymt2D Й. xi ▒ Tanh@xD;
Heliminate higher termsL
w2 = Expand@Simplify@w2 Й. 1 + Tanh@xD ▒ 0DD;
Hasymptotic
behavior for Sech@D and Tanh@DL
w2 = w2 Й. 8Sech@xD ▒ Exp@xD,
1 Tanh@xD ▒ Exp@2 xD<;
w2 = w2 Й. rule1;
w2 = w2 Й. rule2;
w2a = PowerExpand@w2D;
Hdetermine the
coefficients a@kD and b@kDL
akh = Coefficient@w2a, Exp@I k xDD Й. ak ▒ 1;
bkh = Coefficient@w2a, Exp@I k xDD Й. ak ▒ 1 Й akh;
Hcalculate the transmission and
reflection coefficientLTransmission =
1 Й Hakh Conjugate@akhDL Й. k ▒ kin;
Reflection = bkh Conjugate@bkhD Й. k ▒ kin;
8Transmission, Reflection<D;
Hgraphical representation of the
reflection and transmission coefficientL
PlotPT@kini_, kend_, type_D :=
697
698
5.9 Packages and Programs
Block@8k0 = kini, ke = kend, p, t1, label<,
t1 = Transpose@Table@AsymptoticPT@indexxN, kkD,
8kk, k0, ke, Hke k0L Й 5<DD;
If@type m "r", p = t1@@2DD;
label = "╩b╩2 ",
p = t1@@1DD;
label = "╩a╩2 "D;
Plot@Chop@pD, 8indexxN, 1, 2<, AxesLabel ▒
8"N", label<, Prolog ▒ Thickness@0.001DDD;
End@D;
EndPackage@D;
5.9.4 Package CentralField
CentralField is a package allowing you to represent the eigenfunctions for
problems with a central field.
BeginPackage@"CentralField`"D;
Clear@Radial, Angle, AnglePlot, OrbitalD;
Radial::usage = "Radial@ro_, n_, l_,
Z_D calculates the radial representation
of the eigenfunctions for an electron in
the Coulomb potential. The numbers
n and l are the quantum numbers for the
energy and the angular momentum
operator. Z specifies the number of
charges in the nucleus. The radial
distance between the center and the
electron is given by ro.";
Angle::usage = "Angle@theta_, phi_,
l_, m_D calculates the angular part of
the wave function for an electron in the
Coulomb potential. The numbers L
and m denote the quantum numbers for the
angular momentum operator. Theta
and phi are the angles in the spherical
coordinate system.";
5. Quantum Mechanics
Orbital::usage = "Orbital@theta_,
phi_,l_,m_,type_StringD calculates the
superposition of two wave functions for
the quantum numbers m_l = +m and
m_l = m. The variable type allows the
creation of the sum or the difference
of the wave functions. The string values
of type are either plus or minus.";
AnglePlot::usage =
"AnglePlot@pl_,theta_,phi_D gives a graphical
representation of the function contained
in pl. The range of representation
is Pi <= phi < 5 PiЙ2 and 0 < theta <
Pi. Theta is measured with
respect to the vertical axis. This function
is useful for ploting the orbitals
and the angular part of the eigenfunction.";
Hdefine global variablesL
theta::usage;
phi::usage;
ro::usage;
n::usage;
l::usage;
m::usage;
Begin@"`Private`"D;
Hradial part of the eigenfunctions
in the Coulomb potentialL
Radial@ro_, n_, l_, Z_D := Block@8norm, hnl<,
HnormalizationL
norm = HSqrt@Hn + lL ! Й H2 n Hn l 1L !LD
HH2 ZL Й nL ^ Hl + 3 Й 2LL Й H2 l + 1L !;
Hdefinition of the wave functionL
hnl = norm ro ^ l Exp@HHZ roL Й nLD
Hypergeometric1F1@l + 1 n, 2 l + 2, H2 Z roL Й nDD;
699
700
5.9 Packages and Programs
Hangular part of the
eigenfunctions in the Coulomb fieldL
Angle@theta_, phi_, l_, m_D := Block@
8norm, legendre, x, angle, m1, result<, m1 = Abs@mD;
HnormalizationL
norm = H1L ^ m1 Sqrt@
HH2 l + 1L Hl m1L !L Й H2 Hl + m1L !LD Й Sqrt@2 PiD;
HeigenfunctionsL
legendre =
Sin@thetaD ^ m1 D@LegendreP@l, xD, 8x, m1<D;
legendre = legendre Й. x ▒ Cos@thetaD;
Hconsider the cases m>0 and m<0L
If@m √ 0, angle = Exp@I m phiD,
angle = H1L ^ m1 Exp@HI m1 phiLDD;
Hnormalized eigenfunctionL
result = norm legendre angleD;
Hcreate orbitalsL
Orbital@theta_, phi_, l_, m_, type_StringD :=
Block@8norm, m1, rule, wave, wave2<,
m1 = Abs@mD;
Hreplacement rule
for the exponential functionL
rule = 8E ^ HComplex@0, a_D Hx_.LL ▒
Cos@x Abs@aDD + I Sign@aD Sin@x Abs@aDD<;
Hdistinguish different casesL
If@m1 √ 1,
If@type m "plus",
Hsum of the
wave functions for a fixed mL
wave = Expand@Angle@theta, phi, l, m1D +
Angle@theta, phi, l, m1D Й. ruleD,
Hdifference of the wave function
for a fixed mL
wave = Expand@Angle@theta, phi, l, m1D Angle@theta, phi, l, m1D Й. ruleDD;
wave2 = wave ^ 2;
Hnormalization of the superpositionL
norm =
Integrate@wave2, 8phi, 0, 2 Pi<, 8theta, 0, Pi<D;
5. Quantum Mechanics
701
wave2 = Expand@wave2 Й Abs@normDD,
wave = Angle@theta, phi, l, m1D ^ 2DD;
Hgraphical representation
of the angular partL
AnglePlot@pl_, theta_, phi_D := Block@8<,
Htheta is measured with respect to
the vertical axisLParametricPlot3D@
8pl Sin@thetaD Cos@phiD, pl Sin@thetaD Sin@phiD,
pl Cos@thetaD<, 8phi, Pi, 5 Pi Й 2<, 8theta, 0, Pi<,
PlotRange ▒ All, PlotPoints ▒ 840, 40<DD;
End@D;
EndPackage@D;
6
General Relativity
6.1 Introduction
This chapter collects a few examples discussed in connection with general
relativity. The examples are the bending of a light beam in a gravitational
field, Einstein's field equations, the Schwarzschild solution, and the
Reissner?Nordstrom solution for a charged mass point. The given
examples are prominent examples to exemplify the use and techniques of
symbolic computing in the field of general relativity.
General relativity is a widespread theory which today incorporates
different disciplines such as experimental test, exact solutions, formalism
of general relativity, gravitational radiation, gravitational collapses, initial
value problem, alternative theories, unified field theories, quantum gravity,
and many others. In our discussions, we will restrict ourselves to exact
solutions and modeling of gravitational effects. These branches were
originally created by different people. The core contributions were made
by Einstein (see Figure 6.1.1) who based his theory on Riemann's theory
on curved space. The specific contributions of original and successful
704
6.1 Introduction
solutions for different problems originating from Einstein's input were
given by Friedman, Schwarzschild, and others. The derivation of solutions
and applications to specific problems continuous until the present.
Figure 6.1.1.
Albert Einstein: born March 14, 1879; died April 18, 1955.
Riemann (see Figure 6.1.2) by himself was never involved in the creation
of general relativity but contributed a theory that supports efficiently and
successfully to describe the phenomenon of gravitation in a contemporary
way. When Riemann established his theory on curved space, the traditional
theory by Newton was used to describe gravitation phenomena. Newtonian
theory provides an outstanding example for a theory which governed many
centuries of science. At the end of the 19th century, it was becoming
increasingly clear that something was fundamentally wrong with the
current theories, but there was considerable reluctance to make any
fundamental changes to them. Instead, a number of artificial assumptions
required the genius of Einstein to overthrow the prejudices of centuries
and demonstrate in a number of simple thought experiments that some of
the most cherished assumptions of Newtonian theory were untenable. This
was the beginning of relativity. Relativity developed in different stages.
First, with Einstein's brilliant papers in 1905, the special theory of
relativity was introduced. Later, on in the 1920s, Einstein developed
general relativity.
6. General Relativity
Figure 6.1.2.
705
Georg Friedrich Bernhard Riemann: born September 17, 1826; died June 20, 1866.
Out of the general relativity theory a number of old and new questions
arose. One of these questions was the movement of the perihelion of
Mercury. It was an outstanding question of how these movement could be
described in a consistent way. However, Newton's theory allows a way of
explaining how the movement can be motivated, but it remained an open
problem until Einstein's general relativity theory was established. Since
then, many old questions could be attacked. However, there also occurred
new ones due to the mathematics by Riemann. A famous solution of the
spherical Einstein equations was given by Schwarzschild (see Figure
6.1.3). He and others realized that the nonlinear Einstein equations are
very complicated and allow a wealth of new solutions. This will be one of
the subjects in this chapter.
706
6.1 Introduction
Figure 6.1.3.
Karl Schwarzschild: born October 09, 1873; died May 11, 1916.
In Section 6.2 we introduce some notions from general relativity theory.
Light bending is discussed in Section 6.3. Einstein's field equations are
presented in Section 6.4. The Scharzschild solution and the Reissner
Nordstrom solutions are discussed in Sections 6.5 and 6.6.
6. General Relativity
707
6.2 The Orbits in General Relativity
From the classical theory of orbital motion we know that a planet in a
central force field moves in an ellipse around the center of the planetary
system. The orbit of the planet is confined to a plane with fixed
orientation. This behavior is in contradiction to the observations made at
the turn of the century. From observations of the orbital motion of planets,
especially of Mercury, astronomers have discovered that the perihelion of
the orbit is rotating. This movement of the perihelion is called perihelion
shift. The classical theories of Kepler and Newton do not accurately
describe the perihelion shift. The second law of Kepler states that a planet
moves in an ellipse around the center of the planetary system. In classical
theory, the orbital motion is governed by the conservation of energy and
angular momentum. The conservation of angular momentum confines the
planet to a plane. Another conserved quantity of Newton's theory is the
Lenz vector. The Lenz vector is a vector from the focus to the perihelion
that is constant (i.e., in classical theory, the perihelion is at a fixed point in
space). In Einstein's general theory of relativity (GR), these assumptions
are altered. In GR, the orbits are not closed paths and there exists a
perihelion rotation. The actual planetary orbits are rosettes. For these types
of orbit, the perihelion rotates slowly around the Sun. The rotation of the
orbit results from two effects [6.1]:
1. To calculate the orbit using special relativity, we have to take into
account an increase of the mass by
m0
ееееее ,
m = ееееееееееееееее
Х!!!!!!!!!!!!!!!!!!!
(6.2.1)
1-n2 Йc2
where m0 is the rest mass of the planet, c is the velocity of light, and v
is the velocity of the planet in the orbit.
2. The central star produces a gravitational field. According to Einstein's theory, this gravitational field is related to an energy density
which, in turn, is directly connected with a mass density. The additional mass density of the field adds a certain amount of field strength
to the strength of the Sun.
Both effects are relevant in explaining the perihelion shift of a planet. In
the following, we consider the second effect in more detail [6.1]. The Sun
of our solar system possesses a much larger mass than the accompanying
708
6.2 Orbits in General Relativity
planets, which means that we can locate the origin of the coordinate
system in the Sun. Since the orbit is confined to a plane in space
(conservation of angular momentum), we can use plane polar coordinates
Hr, fL to describe the motion of the planets. In GR, the distance between
two points is not simply given by the radial distance r but is also a function
of the radial coordinate. If we denote time by t, we can express the line
element ds2 in space-time in the Schwarzschild metric by
Rs
dr2
ds2 = c2 I1 - ееееrее M dt2 - ееееееее
еееееее - r2 df2 ,
1-Rs Йr
(6.2.2)
Rs
1
ds2 = c2 J1 ccccccc N Dt@tD2 ccccccccccccccccc Dt@rD2 r2 Dt@ID2
Rs
r
1 ccccc
c
r
H? rL2
Rs
- ееееееееееееееее
ееееее + c2 J1 - ееееее еее N H? tL2 - r2 H? fL2
Rs
r
1 - ееее
еr ее
[6.2], where c denotes the speed of light and Rs = 2 G m Й c2 is the
Schwarzschild radius of the gravitational field. G is the gravitational
constant and m is the mass of the Sun. The Lagrangian of the motion in
this metric is given by
Rs
r'2
еееееее - ееее12 r2 f '2 ,
L = c2 I1 - ееееrее M t '2 - ееееееее
1-Rs Йr
(6.2.3)
schwarzschildLagrangian =
H≥s r@sDL2
1
Rs
i
j ccccccccccccc y
z
c2 j1
cccccccccc cccc r@sD2 H≥s I@sDL2
z H≥s t@sDL2 cccccccccccccccc
Rs
2
r@sD {
k
1 ccccccccc
c
r@sD
rё HsL2
Rs
1
- ееееееееееееееее
ееееее е + c2 J1 - ееееееееееее N tё HsL2 - ееееее rHsL2 fё HsL2
Rs
rHsL
2
1 - ееее
еееее
rHsL
6. General Relativity
709
where the primes denote differentiation with respect to the line element s.
Since GR is a geometrically based theory, the orbits of the theory are
derivable by a variational principle. Fermat's principle, which governs the
path of a light beam, is an example from optics. In GR, the orbits follow
from the extremum of the action as determined by the Lagrangian. In close
analogy to our considerations in Section 2.6, the equations of motion of
GR follow from the Euler?Lagrange equations in the form
d
≥L
≥L
ееее
d еsее I ееее
≥r'ее M - ееее
≥rее = 0,
d
≥L
≥L
ееее
еее I ееееееее M - ееее
ее = 0,
d s ≥f'
≥f
(6.2.4)
(6.2.5)
d
≥L
≥L
ееее
еее I ееее
ее M - ееее
ее = 0.
d s ≥t'
≥t
(6.2.6)
swEquations =
EulerLagrange@schwarzschildLagrangian, 8r, I, t<, sD;
swEquations ЙЙ TableForm
Rs r HsL
c Rs t HsL
2 r HsL
- ееееееееееееееее
ееееееееееееее2еее + ееееееееееееееее
еееееееее - rHsL fё HsL2 + ееееееее
ееее
Rsееее Ц 0
Rs 2
rHsL2
ё
2
2
2
ё
ёё
I1- ееее
ее ее M rHsL
rHsL
1- ееее
ее ее
rHsL
fёё HsL rHsL2 + 2 rё HsL fё HsL rHsL Ц 0
2 Rs r HsL t HsL c
2 Rs t HsL c
- ееееееееееееееее
ееееееееееееееееееееее + ееееееееееееееее
ееееееееееее - 2 tёёHsL c2 Ц 0
rHsL
rHsL2
ё
2
ё
ёё
2
Unlike the classical theory of variation, here we consider time t as a
dependent variable. Using Eq. (6.2.3), Eqs. (6.2.5) and (6.2.6) lead to
angular momentum l and energy conservation:
≥L
ееее
еееее = const. = l,
≥ f'
(6.2.7)
≥L
ееее
≥ еt'ее = const. = E0
(6.2.8)
or
1
b
ееее!е ,
r2 f ' = l = ееее
Х!!!!
Rs
(6.2.9)
k2
еееее!ее ,
c2 I1 - ееееrее M t ' = E0 = - ееееееее
Х!!!!
c2 b
(6.2.10)
710
6.2 Orbits in General Relativity
angularMomentum =
1
Map@Integrate@#, sD &, 8swEquationsP2, 1T<DP1T == cccccccccc
Х!!!!
E
1
rHsL2 fё HsL Ц ееееееее
ееее!ее
Х!!!!
b
energy =
MapAt@Integrate@#, sD &, 8swEquationsP3, 1T<, 1DP1T ==
k2
ccccccccccccccccc
Х!!!!
c2 E
k2
2 c2 HrHsL - RsL tё HsL
ееееееееееееееее
еееееееееее Ц - ееееееееееееееее
- ееееееееееееееееееееееееееееееее
ееее!ее
Х!!!!
rHsL
c2 b
where k and b are appropriate constants for the following considerations.
Using the conserved quantities in the expression of the line element
(6.2.2), we get
2
2 4
dr
k r
ееееееее
еееееsее = J- b r4 + ееееееееееееееее
ееееее - r2 N df2 .
R
c2 H1-Rs ЙrL
1- ееее
(6.2.11)
r
Substituting u = 1 Й r simplifies the equation of the orbit to
du 2
k2
ееее M = ееее
ее - H1 - Rs uL Hb + u2 L.
I ееее
df
c2
(6.2.12)
This exact equation is usually solved by using the perturbation theory,
which approximates the solution for a certain range [6.3, 6.4]. In Section
6.8.2, the code is given using the solution steps to solve Eq. (6.2.12). The
package implements the essential steps. Since the equation consists of a
polynomial of third order in u, the solution of Eq. (6.2.12) is expressible
by elliptic functions. To see how this occurs, we carry out the necessary
transformation
4U
1
u = ееееRеsееее + ееее
3 еRеееsе
(6.2.13)
and substitute it into Eq. (6.2.12). The resulting differential equation is the
defining equation for the Weierstrass function 7HzL:
6. General Relativity
711
2
dU
I ееее
еееее M = 4 U 3 - g2 U - g3 .
df
(6.2.14)
However, Mathematica can deliver a preliminary version of this solution
by
DSolve@H≥I U@IDL2 == 4 U@ID3 g2 U@ID g3, U, ID
Solve::tdep : The equations appear to involve the
variables to be solved for in an essentially non-algebraic way. More?
Solve::tdep : The equations appear to involve the
variables to be solved for in an essentially non-algebraic way. More?
:SolveB
jij ij -1 ,
jj2 F jsin H HHRoot@4 #13 - g2 #1 - g3 &, 3D - UHfLL Й HRoot@4 #13 - g2
j k
k
#1 - g3 &, 3D ??
?
Root@4 #13 - g2 #1 - g3 &, 2DLLL ????
??
3
3
Root@4 #1 - g2 #1 - g3 &, 2D - Root@4 #1 - g2 #1 - g3 &, 3D
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееее е
Root@4 #13 - g2 #1 - g3 &, 1D - Root@4 #13 - g2 #1 - g3 &, 3D
yz
z
{
UHfL - Root@4 #13 - g2 #1 - g3 &, 1D
$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
3
Root@4 #1 - g2 #1 - g3 &, 3D - Root@4 #13 - g2 #1 - g3 &, 1D
UHfL - Root@4 #13 - g2 #1 - g3 &, 2D
$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
Root@4 #13 - g2 #1 - g3 &, 3D - Root@4 #13 - g2 #1 - g3 &, 2D
yz
HUHfL - Root@4 #13 - g2 #1 - g3 &, 3DLzzzz Л
{
ij
UHfL - Root@4 #13 - g2 #1 - g3 &, 3D
jj$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
jj ееееееееееееееееееееееееееееееее
3
3
k Root@4 #1 - g2 #1 - g3 &, 2D - Root@4 #1 - g2 #1 - g3 &, 3D
yz
"###################################################
4 UHfL3 - g2 UHfL - g3 zzzz Ц c1 - f, UHfLF,
{
ij i
,
SolveBjjjj2 F jjsin-1 H HHRoot@4 #13 - g2 #1 - g3 &, 3D - UHfLL Й
k
k
HRoot@4 #13 - g2 #1 - g3 &, 3D ?
712
6.2 Orbits in General Relativity
??
?
Root@4 #13 - g2 #1 - g3 &, 2DLLL ????
??
3
3
Root@4 #1 - g2 #1 - g3 &, 2D - Root@4 #1 - g2 #1 - g3 &, 3D
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееее е
Root@4 #13 - g2 #1 - g3 &, 1D - Root@4 #13 - g2 #1 - g3 &, 3D
yz
z
{
UHfL - Root@4 #13 - g2 #1 - g3 &, 1D
$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
3
Root@4 #1 - g2 #1 - g3 &, 3D - Root@4 #13 - g2 #1 - g3 &, 1D
UHfL - Root@4 #13 - g2 #1 - g3 &, 2D
$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
3
Root@4 #1 - g2 #1 - g3 &, 3D - Root@4 #13 - g2 #1 - g3 &, 2D
yz
HUHfL - Root@4 #13 - g2 #1 - g3 &, 3DLzzzz Л
{
ij
UHfL - Root@4 #13 - g2 #1 - g3 &, 3D
jj$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
jj ееееееееееееееееееееееееееееееее
3
3
k Root@4 #1 - g2 #1 - g3 &, 2D - Root@4 #1 - g2 #1 - g3 &, 3D
zy
"###################################################
4 UHfL3 - g2 UHfL - g3 zzzz Ц f + c1 , UHfLF>
{
where
1
HRs L2 b
g2 = ееее
ее - ееееееее4еееееее ,
12
HRs L2 b
HRs L2 k 2
1
g3 = ееее
еееее - ееееееее
еееееее - ееееееее
еееееееее .
216
24
16 c2
(6.2.15)
(6.2.16)
The solution of U is thus
U = 7Hf + C; g2 , g3 L,
(6.2.17)
where C denotes the integration constant. The orbits are now represented
by the coordinates r and f as:
3 Rs
ееееееееееееееее
еееееееее .
rHfL = ееееееееееееееее
1+12 7Hf+C;g
2 ,g3 L
(6.2.18)
6. General Relativity
713
6.2.1 Quasielliptic Orbits
If g2 and g3 are real and the discriminant D = g23 - 27 g32 > 0 we find three
real roots of the characteristic polynomial 4 x3 - g2 x - g3 = 0 which we
call e1 , e2 and e3 . The roots of the characteristic polynomial can be
arranged in the order e2 < e3 < e1 . Using the roots and the expressions g1
and g2 , we can express the periods w1 and w2 of the Weierstrass function
by
╤
w1 = ?
e1
dx
ееееееееееееееее
еееееееееееееееее
Х!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
3
(6.2.19)
4 x -g2 x-g3
and
e2
w2 = i ?
-╤
dx
ееееееееееееееее
еееееееееееееееее .
Х!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
3
(6.2.20)
4 x -g2 x-g3
The first period w1 is a real and the second period w2 is an imaginary
number. w2 is the period of the angle f. If we introduce a third frequency
w3 , the equation of the orbit (6.2.18) is expressible in the form
3 Rs
ееееееееее .
rHfL = ееееееееееееееееееееееееееееееее
1+12 7Hf+w3 ;g2 ,g3 L
(6.2.21)
By introducing w3 , we are able to suppress the singularity of the
Weierstrass function at z = 0. The correct specification of the orbit is made
by the choice of the locations of the perihelion and the aphelion. Choosing
the coordinate system so that the perihelion is reached at f = 0, we get
from Eq. (6.2.21)
3 Rs
3 Rs
ееееееееееее = ееееееее
еееееееее ,
rH0L = ееееееееееееееее
1+12 7H-w3 L
1+12 e3
d r-1
ееееееее
ееее = 0
df
(6.2.22)
(6.2.23)
and
2 -1
d r
ееееееее
еееее < 0.
d f2
(6.2.24)
Once the planet has approached the aphelion, it has traced one-half of the
total orbit. This point of the orbit is characterized by the angle f = w1 . The
radial coordinate at this point is expressed by
s
s
s
3R
3R
3R
еееееееееееееееееее = ееееееееееееееее
ееееееееее = ееееееее
ееееееееее ,
rHw1 L = ееееееееееееееее
1+12 7Hw1 -w3 L
1+12 7Hw2 L
1+12 e2
(6.2.25)
714
6.2 Orbits in General Relativity
-1
dr
ееееееее
d fееее = 0,
(6.2.26)
and
2 -1
d r
ееееееее
еееее > 0.
d f2
(6.2.27)
1
ее + e2 > 0
The relations (6.2.25) and (6.2.27) are correct if the condition ееее
12
2
2
is satisfied. This condition is equivalent to the relation c b > k , relating
the parameters of the Weierstrass function to the physical parameters of
the path. The radial coordinate of the orbit varies between the limits of the
perihelion and the aphelion measured from the origin of the coordinate
system. The two extremal values of the orbit are
3 Rs
еееееееее ,
rP = ееееееее
1+12 e3
3 Rs
rA = ееееееее
еееееееее .
1+12 e
2
(6.2.28)
(6.2.29)
The planet is thus confined between two circles with radii rP and rA . The
path itself is an open orbit in the form of a rosette (see Figure 6.2.4, where
only the path is shown). The orbit in Figure 6.2.4 is similar to the classical
orbit of Kepler's theory. Unlike the classical orbit, the GR shows shifts of
the perihelion and the aphelion. From the classical theory of planet motion,
we know that the difference of phase between two complete rotations is
given by f = 2 p. Within GR the difference in the angle is exactly 2 w1 .
The shift in the perihelion is thus determined by
DfP = 2 Hp - w1 L.
(6.2.30)
6. General Relativity
715
TestPlanet
2 ╣ 108
-8 ╣ 108-6 ╣ 108-4 ╣ 108-2 ╣ 108
2 ╣ 108
-2 ╣ 108
-4 ╣ 108
-6 ╣ 108
-8 ╣ 108
Figure 6.2.4.
Perihelion shift for a system of planets with m = 5.6369 ╣ 1033 kg, a = 5.2325 ╣ 10 8 m and
eccentricity e =0.61713. The numeric value of the perihelion shift is calculated to
be DFP = 90122.8''.
The perihelion shift in the solar system is very small and its experimental
observation is very difficult. However, the calculation of Eq. (6.2.30)
needs to be precise in order to determine the exact numerical value of the
perihelion shift. To calculate the shift using the Weierstrass function, we
need an absolute accuracy of 10-8 in the values for 7HzL. In a graphical
representation of the Mercury orbit for example, the shift is invisible. The
observed and calculated shift for Mercury is 43.1'' for 415 cycles
(approximately one century).
The perihelion and the aphelion are determined by relation (6.2.28). The
locations of the perihelion and the aphelion are usually given by the
classical parameters: the major semiaxis a and eccentricity e. If we
combine both parameters of GR and classical theory, we get the relations
for rP and rA :
p
rP = ееее
еееее ,
1+e
(6.2.31)
716
6.2 Orbits in General Relativity
p
еееее ,
rP = ееее
1-e
(6.2.32)
Х!!!!!!!!!!!!!!!!
where p = b2 Й a and e = a2 - b2 К a. Having determined the extreme
points of the orbit, we know the roots of the Weierstrass function 7: e2 and
e3 from relation (6.2.28). The roots are given by
3 Rs
1
ее I1 - ееееrеAееее M,
e2 = - ееее
12
1
3 Rs
e3 = - ееее
ее I1 - ееее
еееее M.
12
r
(6.2.33)
(6.2.34)
P
In terms of the orbit parameters, we find
1
ее J1 e2 = - ееее
12
s
Х!!!!!!!!!!!!!!!
2
2
3R a
a -b
ееееееее
еееее J1 - ееееееее
еaееееееееее NN,
b2
Х!!!!!!!!!!!!!!!
1
3 Rs a
a2 -b2
e3 = - ееее
е
е
J1
ееееееее
е
еее
е
J1
+
ееееееее
еaееееееееее NN.
2
12
b
(6.2.35)
(6.2.36)
The roots of the 7 function have to satisfy the relations
e1 + e2 + e3 = 0,
2 He21 + e22 + e23 L = g2 ,
4 e1 e2 e3 = g3 .
(6.2.37)
(6.2.38)
(6.2.39)
Here, the root e1 becomes
1
I1 e1 = ееее
6
s
3 aR
ееееееее
ееее M.
b2
(6.2.40)
The quantities g2 and g3 are determined by expressions (6.2.15) and
(6.2.16) and satisfy relations (6.2.38) and (6.2.39). We are now able to
determine the energy E0 and the angular momentum l from the orbital
parameters from Eq. (6.2.9) and (6.2.10). The angular momentum and the
energy can be represented by
l=
s
R
ееееееееееееееее
ееееееее#е ,
"################
1
2
ееее
12ее -g2
ееее1ее - ееее1 g -g
ееее
12ее -g2
3
2 $%%%%%%%%%%%%%%%%%%%%%%%%%%%%
54
6 2
%
E0 = - ееее
ееееееееееееееее
1 ееееееееееее .
c
(6.2.41)
(6.2.42)
One problem with using the exact solution theory is the determination of
the angles w1 and w2 when calculating the perihelion shift with
Mathematica. As mentioned earlier, we need a high degree of accuracy in
our calculation to find the right value for Df. If we do the calculations by
simply integrating Eqs. (6.2.19) and (6.2.20), we have a singularity at one
of the endpoints of the integration interval. Since we have no convergent
representation of the integral, the results are very crude. However, we
know from the theory of the Weierstrass functions that the periods are
6. General Relativity
717
expressible by complete elliptic integrals of the first kind. Using the
properties of the elliptic integrals, we can overcome the inaccurate
numerical integration of Mathematica:
KHmL
e1 -e2
K' HmL
KH1-mL
w2 = i ееееееее
еееееееее!е = i ееееееее
еееееееее!е ,
Х!!!!!!!!!!!!!!
Х!!!!!!!!!!!!!!
e1 -e2
e1 -e2
w1 = ееееееее
еееееееее!е ,
Х!!!!!!!!!!!!!!
(6.2.43)
(6.2.44)
where the module m is given by m = He3 - e2 L Й He1 - e2 L, the roots of the
Weierstrass function.
The above considerations are collected in the Mathematica package
PerihelionShift. An example of the application of PerihelionShift` is
given next. Let us first check the contents of the database for the planets
Planets@"List"D
planet
mean radius
eccentricity mass
Mercury
5.791 ╣ 1010
0.206
1.993 ╣ 1030
Venus
1.082 ╣ 1011
0.007
1.993 ╣ 1030
11
0.017
1.993 ╣ 1030
Icarus
11
1.610 ╣ 10
0.827
1.993 ╣ 1030
Mars
2.228 ╣ 1011
0.093
1.993 ╣ 1030
Ceres
11
0.076
1.993 ╣ 1030
11
0.048
1.993 ╣ 1030
Saturn
12
1.427 ╣ 10
0.056
1.993 ╣ 1030
Uranus
2.870 ╣ 1012
0.047
1.993 ╣ 1030
12
0.009
1.993 ╣ 1030
12
Earth
Jupiter
Neptune
1.497 ╣ 10
4.136 ╣ 10
7.780 ╣ 10
4.496 ╣ 10
Pluto
5.910 ╣ 10
0.250
1.993 ╣ 1030
PSR1916
7.020 ╣ 108
0.617
5.637 ╣ 1030
8
0.617
5.637 ╣ 1033
TestPlanet
5.233 ╣ 10
As result, we get a table containing 13 objects. The last planet is
incorporated to visualize the perihelion shift in a plot. This shift can be
calculated and visualized by
718
6.2 Orbits in General Relativity
Orbit@"TestPlanet"D;
TestPlanet
mass
.56369414099999999 e34
minor axes
323780558.91557515
major axes
523270000.00000006
eccentricity .61713130000000005
Perihelion shift = 90122.8 arcs
TestPlanet
2 ╣ 108
-8 ╣ 108-6 ╣ 108-4 ╣ 108-2 ╣ 108
-2 ╣ 108
-4 ╣ 108
-6 ╣ 108
-8 ╣ 108
2 ╣ 108
6. General Relativity
719
6.2.2 Asymptotic Circles
In this subsection, we discuss a limiting case of GR orbits that is closely
related to the classical orbits of the Kepler theory. We assume that the
constants k and b are such that the discriminant D vanishes. For this case,
two of the roots e1 , e2 , and e3 are equal. If we denote the common root by
e, the remaining root takes the value -2 e. For e > 0, the solution of the
orbit equation (6.2.18) is
s
rHfL =
3 R coshHn fL
ееееееееееееееее
еееееееееееее ,
1-8 n2
(6.2.45)
where n2 = 3 e. This solution results in an apogee with f = 0, provided that
8 n2 < 1. This restriction is equivalent to the condition HRs L2 b > ееее14 .
If f increases, the orbit of the planet spirals down to a circle of asymptotic
radius
3 Rs
еееееее .
r = ееееееее
1+4 n2
(6.2.46)
This radius is smaller than the initial distance of the planet from the center
of the planetary system (see Figure 6.2.5). If we choose n so that the
relation 0 < n2 < ееее18 is satisfied, the radius of the asymptotic circle lies
between the limits 3 Rs and 2 Rs . The orbit for such cases is obtained by
function D0Orbit[] defined in the package PerihelionShift`. An example
for the application of this function to the test planet shows the following
line:
720
6.2 Orbits in General Relativity
D0Orbit@"TestPlanet", 3 SD;
TestPlanet
mass
.56369414099999999 e34
minor axes
323780558.91557515
major axes
523270000.00000006
eccentricity .61713130000000005
Perihelion shift = 90122.8 arcs
1 ╣ 108
5 ╣ 107
5 ╣ 107 1 ╣ 1081.5 ╣ 1082 ╣ 1082.5 ╣ 108
-5 ╣ 107
-1 ╣ 108
Figure 6.2.5.
Orbit for a test planet with D=0.
6.3 Light Bending in the Gravitational Field
Einstein's general theory of relativity predicts that a light ray is bent in a
gravitational field. The corresponding equation of motion follows from the
null geodesic condition ds2 = 0 [6.2]. We discuss the bending of a light ray
in the Schwarzschild metric. The equation of motion is given by
u '' + u - ееее32 Rs u2 = 0,
(6.3.47)
6. General Relativity
721
where u = 1 Й r and Rs = 2 G m Й c2 is the Schwarzschild radius of the mass
m. G denotes the gravitational constant and c is the speed of light.
Multiplying Eq. (6.3.47) by u ' = d u Й d f and integrating it with respect to
parameter s we get
s
2
R
k
ееее12 u '2 + ееее12 u2 - ееее
ее u3 = E = ееее
ее ,
2
c2
(6.3.48)
where E and k, the energy and the scaled energy, are appropriately chosen
constants. The substitution u = 4 U Й Rs + 1 Й H3 Rs L transforms equation
(6.3.48) to a standard form of differential equations defining the
Weierstrass function:
dU 2
ееее M = 4 U 3 - g2 U - g3
I ееее
d еf
(6.3.49)
with
1
ее ,
g2 = ееее
12
HRs L2 k 2
1
g3 = ееее
еееее - ееееееее
еееееееее .
216
16 c2
(6.3.50)
(6.3.51)
The solution for the variable U is given by
U = 7Hf + C; g2 , g3 L.
(6.3.52)
The path of the light ray rHfL is
3 Rs
еееееееееееееееееееееееее .
rHfL = ееееееееееееееее
1+12 7Hf+C;g2 ,g3 L
(6.3.53)
The geometrical locations of the planet and the light rays are given in
Figure 6.3.6. Figure 6.3.6 shows that the light ray has a distance R from
the planet if the angle f = 0.
722
6.3 Light Bending
f1
R
m
Figure 6.3.6.
df
Geometry of light bending in the neighborhood of a mass m. The deviation angle f1 follows
from the relations f2 = p - f1 and df = p - 2 f2 = 2 f1 - p.
When f = f1 , the radius (6.3.53) is infinite. The deviation or bending of
the light ray df is determined by the relation
df = 2 f1 - p
(6.3.54)
(see Figure 6.3.6). Since the Schwarzschild radius Rs and the constant
k 2 Й c2 are greater than zero, it follows that the discriminant
D = g23 - 27 g32 > 0 .
The equation rHf = 0L = R gives us the first expression for the
determination of the roots e1 , e2 , and e3 of the characteristic polynomial
4 t3 - g2 t - g3 = 0. If we set f = 0, it follows from Eq. (6.3.53) that
3 Rs
ееееееееееееееееее .
rHf = 0L = R = ееееееееееееееее
1+12 7HC;g2 ,g3 L
(6.3.55)
If we choose the integration constant as the imaginary period of the
Weierstrass function C = -w2 , we get from the condition 7H-w2 L = e2 the
relation
3 Rs
еееееееее
R = ееееееее
1+12 e2
(6.3.56)
1
and thus e2 = - ееее
ее H1 - 3 Rs Й RL. Since g2 is fixed to 1 Й 12 in the light
12
bending problem, the remaining two roots e1 and e3 satisfy
1
g2 = 2 He21 + e22 + e32 L = ееее
ее ,
12
(6.3.57)
6. General Relativity
723
e1 + e2 + e3 = 0.
(6.3.58)
We find, by eliminating e3 = -He1 + e2 L, in Eq. (6.3.57), the relation
1
g = 0,
e21 + e1 e2 + e22 - ееее
4 2
(6.3.59)
which has the solution
Х!!!!
1
e1 = - ееее
e ■ ееее12
е3ееее "###################
1 - 36 e22# .
2 2
(6.3.60)
From Eq. (6.3.58), we can derive the solution for e1 to be
Х!!!!
1
e ■ ееее12
е3ееее "###################
1 - 36 e22# N.
e3 = -He1 + e2 L = -J ееее
2 2
(6.3.61)
The remaining problem is to find the angle of inclination (i.e., the angle f1
for which the radius tends to infinity). We can express this condition by
3 Rs
еееее .
rHf = f1 L = ╤ = ееееееееееееееееееееееееееееееее
1+12 7Hf1 -w2 ;gееееееее
2 ,g3 L
(6.3.62)
Equation (6.3.62) is satisfied if
1
ее = 0.
7Hf1 - w2 ; g2 , g3 L + ееее
12
(6.3.63)
The frequency w2 is derived from the roots e1 , e2 , and e3 and satisfies the
relations
w2 = w + w',
w1 = w,
w3 = w',
(6.3.64)
(6.3.65)
(6.3.66)
real,
imaginary.
In addition, there are two relations for the frequencies w and w':
KHmL
e1 -e3
еееееееее!е
w = ееееееее
Х!!!!!!!!!!!!!!
and
KH1-mL
e1 -e3
w ' = i ееееееее
еееееееее!е ,
Х!!!!!!!!!!!!!!
(6.3.67)
where the modulus m = He2 - e3 L Й He1 - e3 L. Equation (6.3.63) is only
solvable numerically and provides us with the limiting angle f1 . The angle
determines the asymptotic direction of the light ray. An example of the
bending of a light ray near the surface of the Sun is shown in Figure 6.3.7.
The graphical representation of the light bending is created using Orbit[],
a function of the package LightBending` which is available in Section
6.8.3. The function Deviation[], which is also contained in this package,
allows the numerical calculation of the bending angle. The arguments of
Deviation[] are the mass of the planet and the closest approach of the light
ray.
724
6.3 Light Bending
Orbit@RadiusOfTheSun, MassOfTheSunD;
Figure 6.3.7.
Path of a light ray in the neighborhood of the sun.
The deviation of a light beam passing the Sun can be determined by
Deviation@RadiusOfTheSun, MassOfTheSunD
FindRoot::lstol :
The line search decreased the step size to within tolerance specified by AccuracyGoal and
PrecisionGoal but was unable to find a sufficient decrease in the merit function. You
may need more than 34. digits of working precision to meet these tolerances. More?
Deviation = 1.74416 arcs
8.455905338175976 ╣ 10-6
6. General Relativity
725
6.4 Einstein's Field Equations (Vacuum Case)
Einstein's theory of gravitation can be described by Riemannian geometry.
In Riemannian geometry, space is characterized by its metric. The metric
is normally represented by its line element ds2 or equivalently by the
metric tensor which can be read from the line element. The metric tensor
allows the calculation of the scalar product of two vectors as well as the
equations of motion. Einstein's field equations are the central equations of
GR and describe the motion of a particle in space time. Since GR is
primarily based on geometry, we have to consider the related metric of the
space in addition to the physical problem. For our considerations, we
assume that the independent variables in the space are given by
IndepVar={t,x,y,z}
8t, x, y, z<
The coordinates are used in the determination of the metric tensor. The
function metric[] calculates the coefficients of the metric tensor from a
given line element. metric[] takes the line element ds2 and a list of
coordinates as input variables. The result is the symmetric metric tensor of
the underlying space. The following lines determine the metric tensor the
comments in the function give a short description of the step performed:
metric[lineelement_,independentvars_List]:=Block[
{lenindependent,differentials,diffmatrix,
metricform,varmetric,gh,sum,equation,rule,
varhelp,zeros,zerorule},
(* --- determine the number of independent variables
---*)
lenindependent = Length[independentvars];
(* --- create the differentials corresponding to
dx,dt .... --- *)
differentials = Map[Dt,independentvars];
(* --- a matrix of differential products --- *)
726
6.4 Einstein's Field Equations
diffmatrix = Outer[Times,differentials,
differentials];
(* --- the general metric form --- *)
metricform = Array[gh,{lenindependent,
lenindependent}];
varmetric = Variables[metricform];
(* --- built a system of equations to determine
the elements of the metric
---*)
If[Length[metricform] == Length[diffmatrix],
sum = 0;
Do[
Do[
sum = sum +
metricform[[i,j]] diffmatrix[[i,j]],
{j,1,lenindependent}],
{i,1,lenindependent}],
sum = 0
];
(* --- construct the metric tensor --- *)
If[ sum === 0,
Return[sum],
sum = sum - lineelement;
equation = CoefficientList[sum,
differentials]==0;
rule = Solve[equation,varmetric];
metricform = metricform /. rule;
varmetric = Variables[metricform];
(* --- replace the nonzero elements --- *)
varhelp = {};
Do[
If[Not[FreeQ[varmetric[[i]],gh]],
AppendTo[varhelp,varmetric[[i]] ]
],
{i,1,Length[varmetric]}];
zeros = Table[0,{Length[varhelp]}];
SubstRule[x_,y_]:=x->y;
zerorule = Thread[SubstRule[varhelp,zeros]];
metricform = Flatten[metricform /.
zerorule,1];
(* --- make the metricform symmetric --- *)
metricform = Expand[(metricform +
Transpose[metricform])/2]
];
6. General Relativity
727
metricform
];
Off[Solve::svars];
Off1[Solve::svars];
The application of this function to different examples is demonstrated next.
6.4.1 Examples for Metric Tensors
As a first example, we consider a simple metric of a hypothetical
two-dimensional space in x and t coordinates. The Mathematica symbol
Dt[x] expresses the differential dx in line elements.
MatrixForm@metric@t x Dt@tD2 + x Dt@xD2 , 8x, t<DD
x 0 y
jij
zz
0
k tx{
The result is a (2в2) matrix containing the coefficients of the line element.
A simple three-dimensional example is the Euclidean space with the
well-known cartesian metric. The corresponding line element is
ds2 = dx2 + dy2 + dz2 .
In traditional form, we get the metric by
metricHH? xL2 + H? yL2 + H? zL2 , 8x, y, z<L
1 0 0y
jij
z
jj 0 1 0 zzz
jj
zzz
j
k0 0 1{
which is the expected result for the metric tensor. We see that metric[]
extracts the metric tensor from the line element. The information contained
in the metric tensor is of some importance in the derivation of the field
equations.
728
6.4 Einstein's Field Equations
The line element or the metric tensor for Euclidean space changes its form
if we use a different coordinate system (e.g., the transformation from
cartesian coordinates to spherical coordinates). In spherical coordinates,
the metric tensor is given by
MatrixForm@metricHH? rL2 + r2 H? qL2 + r2 H? fL2 sinHqL, 8r, q, f<LD
0
ij 1 0
yz
jj
zz
jj 0 r2
zz
0
jj
zz
j
z
2
k 0 0 r sinHqL {
where r is the radius and f and q are the spherical polar angles.
A nontrivial example in three dimensions characterizing a curved space is
given by the line element ds2 = dr2 + r2 dq2 + dz2 in cylindrical
coordinates r, f, and z. The corresponding metric tensor is
MatrixForm@metricHH? rL2 + H? zL2 + r2 H? fL2 , 8r, f, z<LD
ij 1 0 0 yz
z
jj
jj 0 r2 0 zzz
zz
jj
z
j
k0 0 1{
In four dimensions ? three space dimensions and one time coordinate ? the
space corresponding to Euclidean space in three dimensions is the
Minkowski space. Euclidean space with cartesian coordinates x, y, and z is
extended by an additional time dimension t. Note the sign difference when
distinguishing between the time coordinate and the space-time dimensions.
The line element in x, y, z, and t is given by ds2 = dt2 - dx2 - dy2 - dz2
(speed of light equals unity, c = 1). The corresponding metric tensor of
Minkowski space reads
6. General Relativity
729
MatrixForm@metricHH? tL2 - H? xL2 - H? yL2 - H? zL2 , 8t, x, y, z<LD
ij 1 0 0 0 yz
jj
z
jj 0 -1 0 0 zzz
jj
z
jj 0 0 -1 0 zzz
jj
zz
jj
zz
0
0
0
-1
k
{
The Minkowski space is a trivial solution of Einstein's field equations for
the vacuum case. A time-independent solution of the field equations with
spherical symmetry is the famous Schwarzschild solution. The line element
ds2 in the coordinates t, r, q, and f is
H? rL2
2m y
i
ds2 = -HH? qL2 + H? fL2 sin2 HqLL r2 - дддддддддддддддд
дддддддддд + jj1 - ддддддддддддд zz H? tL2
2m
r {
k
1 - ддддrддддд
H? rL2
2m
H-H? qL2 - H? fL2 sin2 HqLL r2 - ееееееееееееееее
еееееееее + J1 - ееееееееееее N H? tL2
2m
r
1 - ееее
еrееее
The corresponding metric is
erg = metricHds2, 8t, r, q, f<L; MatrixForm@ergD
2m
ij 1 - ееее
еееее
r
jj
jj 0
jj
jj
jjj 0
jj
jj
k 0
0
0
r
ееееееее
ееееее
2 m-r
0
0
-r2
0
0
yz
zz
zz
0
zz
zz
zz
zz
0
zzz
2
2
-r sin HqL {
0
This representation of the line element is a spherically symmetric solution
of the vacuum field equations. The timelike coordinate t can be interpreted
as the world time. The coordinates q and f can be identified as the usual
angles in spherical coordinates.
The above line element ds2 resembles the line element in Euclidean space.
In the following example, the radial coordinate r is transformed so that we
can
write
the
line
element
in
the
isotropic
form
730
6.4 Einstein's Field Equations
ds2 = GHrL dt2 - FHrL Hd r2 + r2 dq2 + r2 sin2 HqL df2 L. The transformation
reads r = rH1 + m Й H2 rLL2 . The corresponding line element of the metric
reads
m
2
дд д M H? tL2
I1 - дддд
4
2r
y
i m
ds3 = дддддддддддддддддддддддддддддддд
дддддддддддддддддддд - jj дддддддддддд + 1zz HHH? qL2 + H? fL2 sin2 HqLL r2 + H? rL2 L
2
m
{
k2r
ддддд + 1M
I дддд
2r
2
m
I1 - ееее
ееее M H? tL2
4
m
2r
ееееееееееееееееееееееееееееееее
ееееееее
ееееееее - J ееееееееее + 1N HHH? qL2 + H? fL2 sin2 HqLL r2 + H? rL2 L
2
m
2r
ееее + 1M
I ееее
2r
and the corresponding metric tensor is
g = metricHds3, 8t, r, q, f<L
m2
4Um
4 U2
99 cccccccccccccccc
cccccccc
c cccccccccccccccc
cccccccc
c + cccccccccccccccc
ccccccccc , 0, 0, 0=,
2
2
Hm + 2 UL
Hm + 2 UL
Hm + 2 UL2
m4
m3
90, ccccccccccccc
c cccccccc3cc 16 U4
2U
m4
m3
90, 0, ccccccccccccc
c ccccccccc
16 U2
2U
3 m2
cccccccc2cc 2U
3 m2
cccccccccc
2
2m
ccccccccc 1, 0, 0=,
U
2 U m U2 , 0=,
Sin@TD2 m4
Sin@TD2 m3
90, 0, 0, cccccccccccccccc
cccccccccccc cccccccccccccccc
cccccccccccc 16 U2
2U
3
cccc Sin@TD2 m2 2 U Sin@TD2 m U2 Sin@TD2 ==
2
Up to now, we have only discussed the line element of the metric and its
related metric tensor. To derive the field equations for the vacuum case in
GR, we have to introduce other tensors. One of the essential quantities
determining the field equations are the Christoffel symbols. These symbols
are related to the metric tensor in a straightforward way.
6. General Relativity
731
6.4.2 The Christoffel Symbols
Every important relation or equation in a Riemannian space can be
expressed in terms of the metric tensor or its partial derivatives. These
expressions are often very complex. The Christoffel symbols are important
expressions for formulating Einstein's field equations and for expressing
the geometric properties of space. The Christoffel symbols contain the
inverse of the metric tensor ginv and partial derivatives of first order with
respect to the coordinates. The Christoffel symbols can be defined by
Christoffel@m_, a_, b_, g_, ginv_D := Block@8n<,
Expand@
Sum@ginv@@m, nDD HD@g@@a, nDD, IndepVar@@bDD D +
D@g@@b, nDD, IndepVar@@aDD D D@g@@a, bDD, IndepVar@@nDD DL,
8n, 1, Length@gD<D Й 2D
D
In mathematical notation, the function Christoffel[] is given by
mn
Gm
H≥b ga n + ≥a gb n - ≥n ga b L.
a,b = g
(6.4.68)
Other important tensors needed to formulate the field equations are usually
expressed in Christoffel symbols. The Christoffel symbols also appear in
equations for metric geodesics (i.e., the equations defining the
parameterized curve of a particle moving in space). In the following, we
define tensors such as the Riemann tensor, the Ricci tensor, and so forth.
6.4.3 The Riemann Tensor
The curvature tensor, also called the Riemann tensor, is defined in terms of
Christoffel symbols by
732
6.4 Einstein's Field Equations
Riemann@a_, b_, c_, d_, g_, ing_D := Block@8<,
Expand@
D@Christoffel@a, b, d, g, ingD, IndepVar@@cDDD D@Christoffel@a, b, c, g, ingD, IndepVar@@dDDD +
Sum@Christoffel@e, b, d, g, ingD
Christoffel@a, e, c, g, ingD,
8e, 1, Length@gD<D Sum@Christoffel@e, b, c, g, ingD
Christoffel@a, e, d, g, ingD,
8e, 1, Length@gD<D
D
D
The Riemann tensor describes the geometric properties of the underlying
space. A flat space contains a Riemann tensor with zero coefficients.
A contraction of the Riemann tensor delivers the Ricci tensor. The Ricci
tensor is a symmetric tensor in the form
Ricci@m_, q_, g_, ing_D := Block@8a<,
Expand@
Sum@Riemann@a, m, a, q, g, ingD,
8a, 1, Length@gD<DDD
Another contraction of the Ricci tensor defines the curvature scalar or
Ricci scalar:
RicciScalar@g_, ing_D := Block@8<,
Expand@Sum@ing@@a, bDD Ricci@a, b, g, ingD,
8a, 1, Length@gD<, 8b, 1, Length@gD<DDD;
Having these tensors available, we can proceed to the derivation of
Einstein's field equations.
6. General Relativity
733
6.4.4 Einstein's Field Equations
Einstein's vacuum equations are expressed by the Ricci tensor and the
Ricci scalar:
Einstein@m_, n_, g_, ing_D :=
RicciScalar@g, ingD g@@m, nDD
Ricci@m, n, g, ingD cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc
2
The function Einstein[] gives the left-hand side of the equations and the
right-hand side is equal to zero. The derived equations are nonlinear
partial differential equations of second order in space and time. In addition
to the field equations, there are four side conditions given by the Bianchi
identities; these identities are a form of energy conservation:
Bianchi@a_, g_, ing_D := Block@ 8<,
Expand@
Sum@ D@Sum@ ing@@n, mDD Einstein@m, a, g.ingD,
8m, 1, Length@gD<D, IndepVar@@nDD D,
8n, 1, Length@gD<D
+ Sum@ Sum@ Christoffel@n, m, n, g, ingD
Sum@ ing@@m, lDD Einstein@l, a, g, ingD,
8l, 1, Length@gD<D, 8m, 1, Length@gD<D,
8n, 1, Length@gD<D
Sum@ Sum@ Christoffel@n, m, a, g, ingD
Sum@ ing@@m, lDD Einstein@l, n, g, ingD,
8l, 1, Length@gD<D, 8m, 1, Length@gD<D,
8n, 1, Length@gD<D D
D;
734
6.4 Einstein's Field Equations
The calculation of the 10 coefficients of the metric tensor g is an
incompletely formulated mathematical problem since there are fewer
equations than unknowns (6 equations with 10 unknowns). Since the
metric tensor is a solution of the field equations, it is apparent that a
coordinate transformation does not change the problem. When choosing a
coordinate system, we are free to introduce gauge conditions. For example,
Gaussian or normal coordinates are often introduced by setting g0 0 = 1
and g0 a = 0 for a= 1, 2, 3.
We now examine some examples for which we can use the functions
defined above. The first is again the three-dimensional flat cartesian space.
6.4.5 The Cartesian Space
The cartesian space in three dimensions is characterized by the line element
dsc = H? xL2 + H? yL2 + H? zL2
H? xL2 + H? yL2 + H? zL2
with the independent variables
IndepVar = 8x, y, z<
8x, y, z<
The metric form of this space is given by
g = metricHdsc, IndepVarL
1 0 0y
jij
z
jj 0 1 0 zzz
jj
zz
j
z
k0 0 1{
6. General Relativity
735
The inverse of the metric tensor follows by
ing = Inverse@gD
1 0 0y
jij
z
jj 0 1 0 zzz
jj
zzz
j
k0 0 1{
which is simply the identity matrix. Then we calculate some of the
Christoffel symbols to see which of them are not equal to zero.
Christoffel@1, 1, 1, g, ingD
0
Christoffel[1,1,1,g,ing]
0
Christoffel[1,2,1,g,ing]
0
Ricci[1,2,g,ing]
0
It is trivial to see that all Christoffel symbols of this metric vanish.
Consequently, the coefficients of the Riemann tensor vanish, too. This fact
is expected because a cartesian space is flat. We now examine the
cartesian space in different coordinate systems.
736
6.4 Einstein's Field Equations
6.4.6 Cartesian Space in Cylindrical Coordinates
The line element of cartesian space with cylindrical coordinates is
expressed by
IndepVar = 8r, I, z<
8r, f, z<
dscy = H? rL2 + H? zL2 + r2 H? fL2
H? rL2 + H? zL2 + r2 H? fL2
The metric tensor is given by
g = metric[dscy,IndepVar]
ij 1 0 0 yz
jj
z
jj 0 r2 0 zzz
jj
zz
j
z
k0 0 1{
and the inverse of the metric tensor is
ing = Inverse[g]
ij 1 0 0 yz
jj
z
jj 0 ееее1ее 0 zzz
jj
zz
r2
jj
zz
0
0
1
k
{
Contrary to the case of the cartesian coordinate system, the Christoffel
symbols do not all vanish.
6. General Relativity
737
Table[Christoffel[i,j,k,g,ing],{i,1,3},{j,1,3},
{k,1,3}]
80, 0, 0< 80, -r, 0< 80, 0, 0< y
zz
jij
z
jj
jj 80, ееее1r , 0< 8 ееее1r , 0, 0< 80, 0, 0< zzz
zzz
jjj
k 80, 0, 0< 80, 0, 0< 80, 0, 0< {
Nevertheless, the Riemann tensor has to be zero for flat cartesian space in
spite of the coordinate transformation:
Table[Riemann[a,b,c,d,g,ing],{a,1,3},{b,1,3},
{c,1,3},{d,1,3}]
ij ij 0
jj jj
jj jj 0
jj jj
jj 0
jjj k
jj
jj i 0
jj jj
jj jj
jj jj 0
jj j
jj 0
jj k
jj
jj i 0
jj j
jj jj
jj jj 0
jj jj
kk0
0 0y i0 0
zz jj
0 0 zzzz jjjj 0 0
z j
0 0{ k0 0
0 0y i0 0
zz jj
0 0 zzzz jjjj 0 0
z j
0 0{ k0 0
0y
zz
0 zzzz
z
0{
0y
zz
0 zzzz
z
0{
0 0y i0 0 0y
zz jj
zz
0 0 zzzz jjjj 0 0 0 zzzz
z j
z
0 0{ k0 0 0{
0 0 0yy
jij
zz
jj 0 0 0 zzz zzz
jj
zzz zzz
j
z
k 0 0 0 { zzzz
zz
ij 0 0 0 yz zzzz
jj
zz zz
jj 0 0 0 zz zz
jj
zz zz
k 0 0 0 { zzzz
zz
ij 0 0 0 yz zzzz
jjj 0 0 0 zzz zzz
jj
zz zz
j
zz
k0 0 0{{
The disappearance of the Riemann tensor in flat cartesian space is
independent of the corresponding coordinate system. To illustrate the
situation, we next examine the Euclidean space in polar coordinates.
6.4.7 Euclidean Space in Polar Coordinates
With the spherical coordinates
IndepVar = {r,T,I}
8r, q, f<
738
6.4 Einstein's Field Equations
the line element and the corresponding metric are given by
dscp = H? rL2 + r2 H? qL2 + r2 H? fL2 sin2 HqL
H? rL2 + r2 H? qL2 + r2 H? fL2 sin2 HqL
g = metric[dscp,IndepVar]
0
ij 1 0
yz
jj
zz
jj 0 r2
zz
0
jj
zz
j
z
2
k 0 0 r2 sin HqL {
The inverse metric tensor is
ing = Inverse[g]
ij 1 0
jj
jj 0 ееее1ее
jj
r2
jj
j
k0 0
yz
zz
0 zzz
zz
z
csc2 HqL z
ееееееее
ееее
е
еее
е
2
{
r
0
The Christoffel symbols read
Table[Christoffel[i,j,k,g,ing],{i,1,3},{j,1,3},{k,1,3
}]
80, -r, 0<
80, 0, -r sin2 HqL< zy
jij 80, 0, 0<
zz
jj
1
1
jj 80, ееее , 0< 8 ееее , 0, 0< 80, 0, -cosHqL sinHqL< zzz
r
r
jj
zz
jj
zz
1
1
80,
0,
ееее
<
80,
0,
cotHqL<
8
ееее
,
cotHqL,
0<
k
{
r
r
As in the previous example, the Christoffel symbols do not vanish and are
now even more complicated. However, again, as expected, the coefficients
of the Riemann tensor are zero:
6. General Relativity
739
Simplify[Table[Riemann[a,b,c,d,g,ing],{a,1,3},{b,1,3}
,{c,1,3},{d,1,3}] ]
0
jij jij
jj jj 0
jj jj
jj j
jj k 0
jj
jj
jj i 0
jj jj
jj jj 0
jj jj
jj j
jj 0
jj k
jj
jj i 0
jj j
jj jj
jj jj 0
jj jj
kk0
0 0y i0 0
zz jj
0 0 zzzz jjjj 0 0
z j
0 0{ k0 0
0 0y i0 0
zz jj
0 0 zzzz jjjj 0 0
z j
0 0{ k0 0
0y
zz
0 zzzz
z
0{
0y
zz
0 zzzz
z
0{
0 0y i0 0 0y
zz jj
zz
0 0 zzzz jjjj 0 0 0 zzzz
z j
z
0 0{ k0 0 0{
0 0 0yy
jij
zz
jj 0 0 0 zzz zzz
jj
zzz zzz
j
z
k 0 0 0 { zzzz
z
0 0 0 y zzzz
jij
zz zz
jj 0 0 0 zz zz
jjj
zzz zzz
k 0 0 0 { zzzz
z
0 0 0 y zzzz
jij
zz zz
jj 0 0 0 zz zz
jj
zz zz
j
z zz
k0 0 0{{
6.5 The Schwarzschild Solution
6.5.1 The Schwarzschild Metric in Eddington?Finkelstein
Form
In this section, we discuss a nontrivial solution of Einstein's field
equations, the famous Schwarzschild metric given in special coordinate
representations. The Schwarzschild solution is a solution of Einstein's field
equations with the highest symmetry (i.e., with spherical symmetry).
In this representation, there are, as usual, a timelike variable t, a variable r
related to distance, and two angle variables q and f.
IndepVar = {t,r,T,I}
8t, r, q, f<
According to the Eddington?Finkelstein line element,
740
6.5 Schwarzschild Solution
dss = -HH? qL2 + H? fL2 sin2 HqLL r2 2m y
H4 mL ? t ? r
i
ij 2 m
y
j ддддддддддддд + 1zz H? rL2 + jj1 - ддддддддддддд zz H? tL2 - дддддддддддддддд
дддддддддддддддддддддд
r {
r
k
{
k r
2m
2m
4 m ?r ?t
H-H? qL2 - H? fL2 sin2 HqLL r2 - J ееееееееееее + 1N H? rL2 + J1 - ееееееееееее N H? tL2 - ееееееееееееееее
ееееееееееееее
r
r
r
The meaning of r and t is different from the standard Schwarzschild
solution. Due to our choice of r, a nondiagonal element between r and t
appears. Here, the diagonal elements of r and t are in a more symmetric
form. Yet, the metric possesses the required symmetries: spherical
symmetry and time independence. This metric is special in that it is regular
at point r = 2 m, whereas the Schwarzschild line element in its standard
form is singular at this point. This metric can be interpreted as an
analytical extension of the standard form in the region 2 m < r < ╤ to the
region 0 < r < ╤. With the metric tensor
g = metric[dss,IndepVar]
2m
2m
- ееее
еееее
0
0
jij 1 - ееееrеееее
zyz
r
jj
zz
2m
2m
jj - ееее
zz
еееее - ееееrеееее - 1 0
0
jj
zz
r
jj
zz
jj 0
zz
2
0
-r
0
jj
zz
jj
zz
0
0 -r2 sin2 HqL {
k 0
and its inverse
ing = Inverse[g]
Csc@TD2 H2 m r3 Sin@TD2 r4 Sin@TD2 L
2m
99 cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccccccccccc , ccccccccc , 0, 0=,
r
r4
2m
Csc@TD2 H2 m r3 Sin@TD2 + r4 Sin@TD2 L
9 ccccccccc , cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccccccccccc , 0, 0=,
r
r4
Csc@TD2
1
ccccc ==
90, 0, cccccc
c , 0=, 90, 0, 0, cccccccccccccccc
r2
r2
the Christoffel symbols and Ricci tensor are easily calculated.
6. General Relativity
741
Table[Christoffel[i,j,k,g,ing],{i,1,4},{j,1,4},{k,1,4
}]
2 m2
2 m2
m
999 cccccccc
cc , cccccccc
cc + cccccc
c , 0, 0=,
r3
r3
r2
2
2
m
2m
2m
2m
cc + cccccc
c , cccccccc
cc + cccccccc
c , 0, 0=,
9 cccccccc
r3
r2
r3
r2
80, 0, 2 m, 0<, 80, 0, 0, 2 m Sin@TD2 <=,
m
2 m2
2 m2
2 m2
m
2 m2
cc + cccccc
c , cccccccc
cc , 0, 0=, 9 cccccccc
cc , cccccccc
cc cccccc
c , 0, 0=,
99 cccccccc
3
2
3
3
r
r
r
r
r3
r2
80, 0, 2 m r, 0<, 80, 0, 0, 2 m Sin@TD2 r Sin@TD2 <=,
1
1
980, 0, 0, 0<, 90, 0, cccc , 0=, 90, cccc , 0, 0=,
r
r
80, 0, 0, Cos@TD Sin@TD<=, 980, 0, 0, 0<,
1
1
90, 0, 0, cccc =, 80, 0, 0, Cot@TD<, 90, cccc , Cot@TD, 0===
r
r
Table[Ricci[i,j,g,ing],{i,1,4},{j,1,4}]
ij 0
jj
jj 0
jj
jj
jj 0
jj
k0
yz
zz
zz
zz
z
2
2
0 -cot HqL + csc HqL - 1 0 zzzz
z
0
0
0{
0
0
0
0
0
0
With these quantities in hand, we can verify that the form of the
Eddington?Finkelstein line element is a solution of Einstein's vacuum field
equations:
Simplify[ Table[Einstein[a,b,g,ing],{a,1,4},{b,1,4}]
]
ij 0
jj
jj 0
jj
jj 0
jj
jj
k0
0
0
0
0
0
0
0
0
0
0
0
0
yz
zz
zz
zz
zz
zz
zz
{
In addition to the field equations, the Bianchi identities are satisfied also.
742
6.5 Schwarzschild Solution
6.5.2 Dingle's Metric
The metric of Dingle with three space coordinates and one timelike
coordinate
IndepVar = {t,x,y,z}
8t, x, y, z<
is the most general metric in diagonal form.
dsd = A1Ht, x, y, zL H? tL2 - B1Ht, x, y, zL H? xL2 C1Ht, x, y, zL H? yL2 - D1Ht, x, y, zL H? zL2
A1Ht, x, y, zL H? tL2 - B1Ht, x, y, zL H? xL2 C1Ht, x, y, zL H? yL2 - D1Ht, x, y, zL H? zL2
Hence, the metric tensor is a diagonal tensor
g = metric[dsd,IndepVar]
0
0
0
ij A1Ht, x, y, zL
yz
jj
zz
jj
zz
0
-B1Ht,
x,
y,
zL
0
0
zz
jjj
zz
jj
0
0
-C1Ht,
x,
y,
zL
0
zz
jj
zz
j
0
0
0
-D1Ht,
x,
y,
zL
k
{
and so is its inverse
6. General Relativity
743
ing = Inverse[g]
1
ij ееееееееееееееее
еееееееее
0
0
0
jj A1Ht,x,y,zL
jj
1
jj
0
- ееееееееееееееее
ееееееее
0
0
jj
B1Ht,x,y,zL
jj
jj
1
ееееее е
0
0
0
- ееееееееееееееее
jj
C1Ht,x,y,zL
jj
jj
1
0
0
0
- ееееееееееееееее
еееееееее
D1Ht,x,y,zL
k
yz
zz
zz
zz
zz
zz
zz
zz
zz
zz
{
Due to the form of the metric tensor, the Christoffel symbols are fairly
simple expressions.
Table[Christoffel[i,j,k,g,ing],{i,1,4},{j,1,4},{k,1,4
}]
A1H1,0,0,0L @t, x, y, zD
A1H0,1,0,0L @t, x, y, zD
999 cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc ,
2 A1@t, x, y, zD
2 A1@t, x, y, zD
A1H0,0,1,0L @t, x, y, zD
A1H0,0,0,1L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc =,
2 A1@t, x, y, zD
2 A1@t, x, y, zD
A1H0,1,0,0L @t, x, y, zD
B1H1,0,0,0L @t, x, y, zD
9 cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , 0, 0=,
2 A1@t, x, y, zD
2 A1@t, x, y, zD
A1H0,0,1,0L @t, x, y, zD
C1H1,0,0,0L @t, x, y, zD
9 cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , 0, cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , 0=,
2 A1@t, x, y, zD
2 A1@t, x, y, zD
A1H0,0,0,1L @t, x, y, zD
D1H1,0,0,0L @t, x, y, zD
9 cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , 0, 0, cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc ==,
2 A1@t, x, y, zD
2 A1@t, x, y, zD
A1H0,1,0,0L @t, x, y, zD
B1H1,0,0,0L @t, x, y, zD
99 cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , 0, 0=,
2 B1@t, x, y, zD
2 B1@t, x, y, zD
B1H1,0,0,0L @t, x, y, zD
B1H0,1,0,0L @t, x, y, zD
9 cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc ,
2 B1@t, x, y, zD
2 B1@t, x, y, zD
B1H0,0,1,0L @t, x, y, zD
B1H0,0,0,1L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc =,
2 B1@t, x, y, zD
2 B1@t, x, y, zD
B1H0,0,1,0L @t, x, y, zD
C1H0,1,0,0L @t, x, y, zD
90, cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , 0=,
2 B1@t, x, y, zD
2 B1@t, x, y, zD
B1H0,0,0,1L @t, x, y, zD
90, cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc ,
2 B1@t, x, y, zD
D1H0,1,0,0L @t, x, y, zD
0, cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc ==,
2 B1@t, x, y, zD
A1H0,0,1,0L @t, x, y, zD
C1H1,0,0,0L @t, x, y, zD
99 cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , 0, cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , 0=,
2 C1@t, x, y, zD
2 C1@t, x, y, zD
744
6.5 Schwarzschild Solution
B1H0,0,1,0L @t, x, y, zD
C1H0,1,0,0L @t, x, y, zD
90, cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , 0=,
2 C1@t, x, y, zD
2 C1@t, x, y, zD
C1H1,0,0,0L @t, x, y, zD
C1H0,1,0,0L @t, x, y, zD
9 cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc ,
2 C1@t, x, y, zD
2 C1@t, x, y, zD
C1H0,0,1,0L @t, x, y, zD
C1H0,0,0,1L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc =, 90,
2 C1@t, x, y, zD
2 C1@t, x, y, zD
C1H0,0,0,1L @t, x, y, zD
D1H0,0,1,0L @t, x, y, zD
0, cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc ==,
2 C1@t, x, y, zD
2 C1@t, x, y, zD
A1H0,0,0,1L @t, x, y, zD
D1H1,0,0,0L @t, x, y, zD
99 cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , 0, 0, cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc =,
2 D1@t, x, y, zD
2 D1@t, x, y, zD
B1H0,0,0,1L @t, x, y, zD
D1H0,1,0,0L @t, x, y, zD
90, cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , 0, cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc =,
2 D1@t, x, y, zD
2 D1@t, x, y, zD
C1H0,0,0,1L @t, x, y, zD
D1H0,0,1,0L @t, x, y, zD
90, 0, cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc =,
2 D1@t, x, y, zD
2 D1@t, x, y, zD
D1H1,0,0,0L @t, x, y, zD
D1H0,1,0,0L @t, x, y, zD
9 cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc ,
2 D1@t, x, y, zD
2 D1@t, x, y, zD
D1H0,0,1,0L @t, x, y, zD
D1H0,0,0,1L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc ===
2 D1@t, x, y, zD
2 D1@t, x, y, zD
Still, one equation of Einstein's vacuum field equations is complicated
Einstein[1,1,g,ing]
A1H0,0,0,1L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc +
4 A1@t, x, y, zD D1@t, x, y, zD
2
A1H0,0,0,1L @t, x, y, zD B1H0,0,0,1L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc +
4 B1@t, x, y, zD D1@t, x, y, zD
A1H0,0,0,1L @t, x, y, zD C1H0,0,0,1L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc 4 C1@t, x, y, zD D1@t, x, y, zD
A1H0,0,0,1L @t, x, y, zD D1H0,0,0,1L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc +
4 D1@t, x, y, zD2
A1H0,0,1,0L @t, x, y, zD
A1H0,0,0,2L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccc +
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc cccccccccccccccccccccccccccccccc
4 A1@t, x, y, zD C1@t, x, y, zD
2 D1@t, x, y, zD
2
A1H0,0,1,0L @t, x, y, zD B1H0,0,1,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc 4 B1@t, x, y, zD C1@t, x, y, zD
A1H0,0,1,0L @t, x, y, zD C1H0,0,1,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc +
4 C1@t, x, y, zD2
A1H0,0,1,0L @t, x, y, zD D1H0,0,1,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc +
4 C1@t, x, y, zD D1@t, x, y, zD
6. General Relativity
745
A1H0,0,2,0L @t, x, y, zD
A1H0,1,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc 2 C1@t, x, y, zD
4 A1@t, x, y, zD B1@t, x, y, zD
2
A1H0,1,0,0L @t, x, y, zD B1H0,1,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc +
4 B1@t, x, y, zD2
A1H0,1,0,0L @t, x, y, zD C1H0,1,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc +
4 B1@t, x, y, zD C1@t, x, y, zD
A1H0,1,0,0L @t, x, y, zD D1H0,1,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc +
4 B1@t, x, y, zD D1@t, x, y, zD
A1H0,2,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc +
2 B1@t, x, y, zD
A1H1,0,0,0L @t, x, y, zD B1H1,0,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc +
4 A1@t, x, y, zD B1@t, x, y, zD
B1H1,0,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccccc +
4 B1@t, x, y, zD2
2
A1H1,0,0,0L @t, x, y, zD C1H1,0,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc +
4 A1@t, x, y, zD C1@t, x, y, zD
C1H1,0,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccccc +
4 C1@t, x, y, zD2
2
A1H1,0,0,0L @t, x, y, zD D1H1,0,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc +
4 A1@t, x, y, zD D1@t, x, y, zD
B1H2,0,0,0L @t, x, y, zD
D1H1,0,0,0L @t, x, y, zD
cccccccccccccccc
ccccccccc cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccc
ccc cccccccccccccccccccccccccccccccc
2
2 B1@t, x, y, zD
4 D1@t, x, y, zD
2
C1H2,0,0,0L @t, x, y, zD
D1H2,0,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc 2 C1@t, x, y, zD
2 D1@t, x, y, zD
2
i
1
A1H0,0,0,1L @t, x, y, zD
j
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccc
cc +
cccc A1@t, x, y, zD j
j
j 2 A1@t, x, y, zD2 D1@t, x, y,cccccccc
2
zD
k
A1H0,0,0,1L @t, x, y, zD B1H0,0,0,1L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccc 2 A1@t, x, y, zD B1@t, x, y, zD D1@t, x, y, zD
B1H0,0,0,1L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccc +
2 B1@t, x, y, zD2 D1@t, x, y, zD
2
A1H0,0,0,1L @t, x, y, zD C1H0,0,0,1L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccc +
2 A1@t, x, y, zD C1@t, x, y, zD D1@t, x, y, zD
B1H0,0,0,1L @t, x, y, zD C1H0,0,0,1L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccc 2 B1@t, x, y, zD C1@t, x, y, zD D1@t, x, y, zD
C1H0,0,0,1L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccc 2 C1@t, x, y, zD2 D1@t, x, y, zD
2
A1H0,0,0,1L @t, x, y, zD D1H0,0,0,1L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc 2 A1@t, x, y, zD D1@t, x, y, zD2
B1H0,0,0,1L @t, x, y, zD D1H0,0,0,1L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc 2 B1@t, x, y, zD D1@t, x, y, zD2
746
6.5 Schwarzschild Solution
C1H0,0,0,1L @t, x, y, zD D1H0,0,0,1L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc +
2 C1@t, x, y, zD D1@t, x, y, zD2
A1H0,0,0,2L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccc +
A1@t, x, y, zD D1@t, x, y, zD
B1H0,0,0,2L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccc +
B1@t, x, y, zD D1@t, x, y, zD
C1H0,0,0,2L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccc C1@t, x, y, zD D1@t, x, y, zD
A1H0,0,1,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccc +
2 A1@t, x, y, zD2 C1@t, x, y, zD
2
A1H0,0,1,0L @t, x, y, zD B1H0,0,1,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccc 2 A1@t, x, y, zD B1@t, x, y, zD C1@t, x, y, zD
B1H0,0,1,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccc 2 B1@t, x, y, zD2 C1@t, x, y, zD
2
A1H0,0,1,0L @t, x, y, zD C1H0,0,1,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc 2 A1@t, x, y, zD C1@t, x, y, zD2
B1H0,0,1,0L @t, x, y, zD C1H0,0,1,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc +
2 B1@t, x, y, zD C1@t, x, y, zD2
A1H0,0,1,0L @t, x, y, zD D1H0,0,1,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccc +
2 A1@t, x, y, zD C1@t, x, y, zD D1@t, x, y, zD
B1H0,0,1,0L @t, x, y, zD D1H0,0,1,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccc 2 B1@t, x, y, zD C1@t, x, y, zD D1@t, x, y, zD
C1H0,0,1,0L @t, x, y, zD D1H0,0,1,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc 2 C1@t, x, y, zD2 D1@t, x, y, zD
D1H0,0,1,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccc +
2 C1@t, x, y, zD D1@t, x, y, zD2
2
A1H0,0,2,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccc +
A1@t, x, y, zD C1@t, x, y, zD
B1H0,0,2,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccc +
B1@t, x, y, zD C1@t, x, y, zD
D1H0,0,2,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccc C1@t, x, y, zD D1@t, x, y, zD
A1H0,1,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccc 2 A1@t, x, y, zD2 B1@t, x, y, zD
2
A1H0,1,0,0L @t, x, y, zD B1H0,1,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc +
2 A1@t, x, y, zD B1@t, x, y, zD2
A1H0,1,0,0L @t, x, y, zD C1H0,1,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccc 2 A1@t, x, y, zD B1@t, x, y, zD C1@t, x, y, zD
B1H0,1,0,0L @t, x, y, zD C1H0,1,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc 2 B1@t, x, y, zD2 C1@t, x, y, zD
6. General Relativity
747
C1H0,1,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccc +
2 B1@t, x, y, zD C1@t, x, y, zD2
2
A1H0,1,0,0L @t, x, y, zD D1H0,1,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccc 2 A1@t, x, y, zD B1@t, x, y, zD D1@t, x, y, zD
B1H0,1,0,0L @t, x, y, zD D1H0,1,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc +
2 B1@t, x, y, zD2 D1@t, x, y, zD
C1H0,1,0,0L @t, x, y, zD D1H0,1,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccc 2 B1@t, x, y, zD C1@t, x, y, zD D1@t, x, y, zD
D1H0,1,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccc +
2 B1@t, x, y, zD D1@t, x, y, zD2
2
A1H0,2,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccc +
A1@t, x, y, zD B1@t, x, y, zD
C1H0,2,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccc +
B1@t, x, y, zD C1@t, x, y, zD
D1H0,2,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccc +
B1@t, x, y, zD D1@t, x, y, zD
A1H1,0,0,0L @t, x, y, zD B1H1,0,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc +
2 A1@t, x, y, zD2 B1@t, x, y, zD
B1H1,0,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccc +
2 A1@t, x, y, zD B1@t, x, y, zD2
2
A1H1,0,0,0L @t, x, y, zD C1H1,0,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc 2 A1@t, x, y, zD2 C1@t, x, y, zD
B1H1,0,0,0L @t, x, y, zD C1H1,0,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccc +
2 A1@t, x, y, zD B1@t, x, y, zD C1@t, x, y, zD
C1H1,0,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccc +
2 A1@t, x, y, zD C1@t, x, y, zD2
2
A1H1,0,0,0L @t, x, y, zD D1H1,0,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc 2 A1@t, x, y, zD2 D1@t, x, y, zD
B1H1,0,0,0L @t, x, y, zD D1H1,0,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccc 2 A1@t, x, y, zD B1@t, x, y, zD D1@t, x, y, zD
C1H1,0,0,0L @t, x, y, zD D1H1,0,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccc +
2 A1@t, x, y, zD C1@t, x, y, zD D1@t, x, y, zD
D1H1,0,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccc 2 A1@t, x, y, zD D1@t, x, y, zD2
2
B1H2,0,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccc A1@t, x, y, zD B1@t, x, y, zD
C1H2,0,0,0L @t, x, y, zD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccc A1@t, x, y, zD C1@t, x, y, zD
y
D1H2,0,0,0L @t, x, y, zD
z
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccc z
z
A1@t, x, y, zD D1@t, x, y, zD z
{
748
6.5 Schwarzschild Solution
6.5.3 Schwarzschild Metric in Kruskal Coordinates
The Kruskal solution is the most general analytical extension of the
Schwarzschild metric. Whereas the Eddington?Finkelstein solution is
developed for the time region 0 ╖ t < ╤ or -╤ < t ╖ 0, the Kruskal
solution is extended to both time regions.
The Kruskal solution consists of the two angle variables q and f, a
spacelike variable x and a timelike variable t.
IndepVar = {t,x,T,I}
8t, x, q, f<
The radial distance r is defined implicitly by the equation
rHx,tL
дддддд
2m
gld = t2 - x2 == -HrHx, tL - 2 mL ? дддддддд
rHx,tL
ееее е
2m
t2 - x2 Ц ? ееееееее
H2 m - rHx, tLL
For later calculations, this equation is solved for t:
seq = Last@Solve@gld Й. rHx, tL ф r, tDD
:t ь
r
"########################################
x2 + ? ееее2еmе е е H2 m - rL >
The line element is given by the radial coordinate r:
6. General Relativity
749
rHx,tL
2
дддддд
2 m H16 m 2 L H? tL
?- дддддддд
dsk = дддддддддддддддддддддддддддддддд
дддддддддддддддд
ддддддддддддддддд r
rHx,tL
2
дддддд
2 m H? xL
H16 m2 L ?- дддддддд
дддддддддддддддддддддддддддддддд
дддддддддддддддд
дддддддддддддддддд - rHx, tL2 HH? qL2 + H? fL2 sin2 HqLL
rHx, tL
rHx,tL
rHx,tL
ееее е 2
ееее е 2
2m
2m
16 ?- ееееееее
m H? tL2
m H? xL2
16 ?- ееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееее - rHx, tL2 HH? qL2 + H? fL2 sin2 HqLL - ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееее
r
rHx, tL
The metric is again in the shape of a diagonal matrix and its inverse
g = metric[dsk,IndepVar]
ij 16 ?- ееееееее2ееееmеееееее m2
ееееееееееееееее
0
0
0
jj ееееееееееееееее
r
jj
jj
rHx,tL
- ееееееее
ееее
е
еее
е
е
е
2
jj
16 ? 2 m m
jj
0
- ееееееееееееееее
ееееееееееееееее
0
0
rHx,tL
jj
jj
2
jj
0
0
-rHx, tL
0
jj
j
0
0
0
-rHx, tL2 sin2 HqL
k
rHx,tL
yz
zz
zz
zz
zz
zz
zz
zz
zz
zz
z
{
ing = Inverse[g]
ij ? ееееееее
2ееееmеееееее r
ееееееееее
0
0
0
jj ееееееее
jj 16 m2
jj
rHx,tL
еееееее
jj
2ееееm
? ееееееее
rHx,tL
jj 0
еееееееееее
0
0
- ееееееееееееееее
16 m2
jj
jj
1
jj 0
0
- ееееееее
еееееее
0
jj
rHx,tL2
jj
jj
csc2 HqL
0
0
0
- ееееееее
ееееее е
rHx,tL2
k
rHx,tL
yz
zz
zz
zz
zz
zz
zz
zz
zz
zz
zz
zz
{
To calculate the Christoffel symbols and the Einstein tensor, we compute
the derivatives of r@x, tD up to second order following from equation gld.
750
6.5 Schwarzschild Solution
s1= Flatten[Simplify[Solve[D[gld,x],D[r[x,t],x]]]];
s2 = Flatten[Simplify[Solve[D[gld,t],D[r[x,t],t]]]];
s3 =
Flatten[Simplify[Solve[D[gld,x,x],D[r[x,t],x,x]]
/.s1 ]];
s4 =
Flatten[Simplify[Solve[D[gld,t,t],D[r[x,t],t,t]] /.
s2 ]];
sg = Flatten[{s1,s2,s3,s4}]
rHx,tL
rHx,tL
ееее е
ееее е
2m
2m
4 ?- ееееееее
4 ?- ееееееее
mx
mt
:rH1,0L Hx, tL ь ееееееееееееееее
еееееееееееееееееееее , rH0,1L Hx, tL ь - ееееееееееееееее
ееееееееееееееееееее ,
rHx, tL
rHx, tL
rHx,tL
H2,0L
r
rHx,tL
ееее е
2m
rHx, tL2 M
4 ?- ееееееееmееее е m I-4 m x2 - 2 rHx, tL x2 + ? ееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееее ,
Hx, tL ь ееееееееееееееееееееееееееееееее
rHx, tL3
rHx,tL
rHx,tL
ееее е
2m
rHx, tL2 M
4 ?- ееееееееmееее е m I4 m t2 + 2 rHx, tL t2 + ? ееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
ееееееееее >
rH0,2L Hx, tL ь - ееееееееееееееееееееееееееееееее
rHx, tL3
With the list of sg rules, the Christoffel symbols and the Einstein tensor are
calculated as follows:
6. General Relativity
751
Table[ Simplify[ Christoffel[i,j,k,g,ing] /. sg
],{i,1,4},{j,1,4},{k,1,4}]
r@x,tD
r@x,tD
ccccccc t
ccccccc x
2m
2m
ф cccccccc
ф cccccccc
999 cccccccccccccccc
ccccccc , cccccccccccccccc
ccccccc , 0, 0=,
r@x, tD
r@x, tD
r@x,tD
r@x,tD
ccccccc r t H2 m + r@x, tDL
2 cmcccccc x
2m
ф cccccccc
ф cccccccc
ccccccc , cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccccccccccc , 0, 0=,
9 cccccccccccccccc
r@x, tD
r@x, tD3
rt
r t Sin@TD2
ccccccccccccc ==,
90, 0, ccccccccc , 0=, 90, 0, 0, cccccccccccccccc
4m
4m
r@x,tD
r@x,tD
ccccccc t H2 m + r@x, tDL
2 cmcccccc x
2m
ф cccccccc
ф cccccccc
99 cccccccccccccccc
ccccccc , cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccccccc , 0, 0=,
r
r@x, tD2
r@x,tD
ccccccc t H2 m + r@x, tDL
2m
ф cccccccc
cccccccccccccccc
ccccccccccccc ,
9 cccccccccccccccccccccccccccccccc
r@x, tD2
r@x,tD
2 cmcccccc x H2 m + r@x, tDL
ф cccccccc
cccccccccccccccc
ccccccccccccc , 0, 0=,
cccccccccccccccccccccccccccccccc
r@x, tD2
x r@x, tD
x r@x, tD Sin@TD2
ccccccccccccc ==,
90, 0, cccccccccccccccc
ccccccccc , 0=, 90, 0, 0, cccccccccccccccccccccccccccccccc
4m
4m
r@x,tD
r@x,tD
2 cmcccccc m t
2 cmcccccc m x
4 ф cccccccc
4 ф cccccccc
990, 0, cccccccccccccccc
cccccccc2ccccccc , 0=, 90, 0, cccccccccccccccc
ccccccccccccccc , 0=,
r@x, tD
r@x, tD2
r@x,tD
r@x,tD
ccccccc m t
ccccccc m x
2m
2m
4 ф cccccccc
4 ф cccccccc
9 cccccccccccccccc
cccccccc2ccccccc , cccccccccccccccc
ccccccccccccccc , 0, 0=,
r@x, tD
r@x, tD2
r@x,tD
ccccccc m t
2m
4 ф cccccccc
ccccccccccccccc =,
80, 0, 0, Cos@TD Sin@TD<=, 990, 0, 0, cccccccccccccccc
r@x, tD2
r@x,tD
ccccccc m x
2m
4 ф cccccccc
ccccccccccccccc =, 80, 0, 0, Cot@TD<,
90, 0, 0, cccccccccccccccc
r@x, tD2
r@x,tD
r@x,tD
ccccccc m t
ccccccc m x
2m
2m
4 ф cccccccc
4 ф cccccccc
9 cccccccccccccccc
cccccccc2ccccccc , cccccccccccccccc
ccccccccccccccc , Cot@TD, 0===
r@x, tD
r@x, tD2
To verify Einstein's field equations, we calculate, for example, the (1,1)
coefficient of the Einstein tensor:
es1 = Simplify[ Einstein[1,1,g,ing] /. sg
rHx,tL
rHx,tL
rHx,tL
ееее е
ееее е
2m
2m
8 ?- ееееееееmееее е m I-r t2 - ? ееееееее
rHx, tL2 + Ix2 + 2 ? ееееееее
mM rHx, tLM
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
ееееееееееееееее
еееееееееееее
3
r rHx, tL
]
752
6.5 Schwarzschild Solution
With the aid of the defining equation for r, the above expression vanishes.
es1 = es1 /. { r[x,t] ▒ r}
r
r
r
8 ?- ееmеее m I-? ееее2еmе е е r2 - t2 r + Ix2 + 2 ? ееее2еmе е е mM rM
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееееееееееееееееееее
ееееееееееееееее
еееееееееееееееееееее
r4
Simplify[ PowerExpand[es1 /. seq ] ]
0
6.6 The Reissner?Nordstrom Solution for a
Charged Mass Point
The Reissner?Nordstrom solution is a spherically symmetric metric for a
massive body with charge ╤. This type of solution allows the study of the
coupling of Einstein's field equations with Maxwell's equations via the
energy momentum tensor. Consequently, we have to solve the
inhomogeneous field equations. Because of the spherical symmetry, we
can use the Kruskal variables:
IndepVar = {t,r,T,I}
8t, r, q, f<
The same shape of the line element is also given:
dsr = -HH? qL2 + H? fL2 sin2 HqLL r2 - ?lHrL H? rL2 + ?nHrL H? tL2
H-H? qL2 - H? fL2 sin2 HqLL r2 - ?lHrL H? rL2 + ?nHrL H? tL2
The metric tensor follows
6. General Relativity
753
g = metric[dsr,IndepVar]
0
0
0
ij ?nHrL
yz
jj
zz
jj 0 -?lHrL 0
zz
0
jj
zz
jj
zz
2
jj 0
zz
0
0
-r
jj
zz
j
z
2
2
0
0
0
-r
sin
HqL
k
{
The related inverse metric tensor is
ing = Inverse[g]
0
0
0
ij ?-nHrL
jj
-lHrL
jj 0
-?
0
0
jj
jj
1
jj 0
ееее
е
е
0
0
jj
r2
jj
j
csc2 HqL
0
0 - ееееееее
ееееее е
k 0
r2
yz
zz
zz
zz
zz
zz
zz
zz
z
{
Since the Reissner?Nordstrom solution possesses spherical symmetry, the
coordinates can be chosen so that the metric is static and n and l depend
only on the radial distance r. At the same time, the Reissner?Nordstrom
solution satisfies Einstein's field equations and Maxwell's vacuum
equations. Consequently, the Maxwell tensor F also depends on the
distance r. Its form is determined by a purely radial electrostatic field.
F = {{ 0, Ee[r],0,0},{Ee[r],0,0,0},{0,0,0,0},{0,0,0,0}}
-EeHrL 0 0 y
ij 0
zz
jj
jj EeHrL
0
0 0 zzzz
jj
z
jj 0
0
0 0 zzzz
jj
jj
zz
0
0 0{
k 0
According to Maxwell's equations, the covariant divergence of the
Maxwell tensor must vanish. The conditions deliver the substitution rule
754
6.6 Reissner Nordstrom Solution
1
╤ ? дд2дд HlHrL+nHrLL
sm = 9EeHrL ф дддддддддддддддд
дддддддддддддддд
дддддддддд =
r2
1
? ее2ее HlHrL+nHrLL ╤
ееееееееееееееее
ееееееее >
:EeHrL ь ееееееееееееееее
r2
and the Maxwell tensor
F = F /. sm
? ее2еее HlHrL+nHrLL ╤
jij
еееееееееееееее
0
- ееееееееееееееее
jj
r2
jj 1
jj ? ее2еее HlHrL+nHrLL ╤
jj ееееееееееееееее
еееееееееееееее
0
jj
r2
jj
jj
0
0
jj
j
0
0
k
1
y
0 0 zzz
zz
zz
z
0 0 zzzz
zz
0 0 zzzz
z
0 0{
with the corresponding covariant tensor.
Fc = Simplify[ing . F . ing ]
ij
0
jj
jj
jj
1
jj ?- ее2еее HlHrL+nHrLL ╤
jj - ееееееееееееееее
еееееееееееееееее
r2
jj
jj
0
jjj
j
0
k
y
?- ее2еее HlHrL+nHrLL ╤
ееееееееееееееее
ееееееееееееееееее 0 0 zzz
r2
zz
zz
z
0
0 0 zzzz
zz
0
0 0 zzzz
z
0
0 0{
1
The energy momentum tensor T is computed by
6. General Relativity
755
T = SimplifyA
4 4
1
i cccccccc
TableA ? ? j
c ing@@c, dDD F@@a, cDD F@@b, dDD +
j
4S
c=1 d=1 k
1
z
cccccccccccc g@@a, bDD F@@c, dDD Fc@@c, dDDy
z,
16 S
{
8a, 1, 4<, 8b, 1, 4<EE
? ╤
еееее4ееее
0
jij ееееееее
jj 8 p r
lHrL ╤2
jj
?
jj 0
еееееееее
- ееееееее
8 p r4
jj
jj
jj 0
0
jj
jj
jj
0
k 0
nHrL 2
0
0
╤
ееееееее
еееее
8 p r2
2
0
zyz
zz
zz
zz
0
zz
zz
zz
0
zz
zz
2
2
z
╤ sin HqL z
ееееееее
ееееееее
ееее
е
е
2
{
8pr
0
It should be pointed out that the energy momentum tensor for a source-free
electromagnetic field is traceless since the Maxwell tensor ? a fully
antisymmetric tensor ? is traceless. According to this property of the
energy momentum tensor, the Ricci scalar vanishes as well. Consequently,
the field equations reduce to R = 8 p T, where R is the Ricci tensor.
Simplify[Table[Ricci[a,b,g,ing] -8 S
T[[a,b]],{a,1,4},{b,1,4}] ]
1
99 cccccccc4cc
4r
HфO@rD+Q@rD H4 фO@rD ≥2 + r3 H4 r O┘ @rDL Q┘ @rD + r4 Q┘ @rD2 +
2 r4 Q┘┘ @rDLL, 0, 0, 0=,
1 j 4 фO@rD ≥2
4
z
90, cccc i
ccccccc Q┘ @rD2 + O┘ @rD J cccc + Q┘ @rDN 2 Q┘┘ @rDy
j cccccccccccccccc
z,
4 k
r4
r
{
0, 0=, 90, 0,
2 ≥2
1 j
z, 0=,
cc + фO@rD r O┘ @rD фO@rD r Q┘ @rDy
cccc i
2 2 фO@rD cccccccc
r2
2 k
{
1
2
O@rD
90, 0, 0, cccccccc2cc Hф
Sin@TD
2r
2
O@rD 2
H2 r + 2 ф
r 2 фO@rD ≥2 + r3 O┘ @rD r3 Q┘ @rDLL==
The solutions of these differential equations can easily be verified. With
the coordinates
6.6 Reissner Nordstrom Solution
756
IndepVar = {t,r,T,I}
8t, r, q, f<
the line element is given by
dsrn =
H? rL2
2m
y
i ╤2
дддддддддддддддд
дддддддддддд + jjjj ддддддддд - дддддддддддд + 1zzzz H? tL2
-HH? qL2 + H? fL2 sin2 HqLL r2 - дддддддддддддддд
2
2
╤
2m
r
r
{
ддддr2дд - ддддrддддд + 1 k
H? rL2
2m
y
i ╤2
H-H? qL2 - H? fL2 sin2 HqLL r2 - ееееееееееееееее
ееееееееееееееее
еееееееее + jj еееее2еее - ееееееееееее + 1zz H? tL2
╤2
2m
r
r
{
k
ееее
е
е
ееее
е
е
ее
+
1
r2
r
and the metric tensor
g = metric[dsrn,IndepVar];g//MatrixForm
╤
2m
yz
ij ееее
0
0
0
zz
jj r2ее - ееееrеееее + 1
zz
jj
2
r
zz
jj
ееееееее
ееее
е
0
0
0
- ееееееееееееееее
zz
jj
2
2
r
-2
m
r+╤
zz
jj
zz
jj
2
0
0
0
-r
zz
jj
zz
jj
2
2
0
0
0
-r
sin
HqL
{
k
2
with the corresponding inverse
ing = Simplify[ Inverse[g] ];ing//MatrixForm
r
ij ееееееееееееееее
ееее
r+╤2
jj r2 -2 mееееееее
jj
╤2
jj
0
- ееее
ее
jj
r2
jj
jj
0
jj
jj
jj
0
k
2
yz
zz
zz
2m
zz
+ ееееrеееее - 1 0
0
zz
zz
zz
1
ее
0
0
- ееее
zz
r2
zz
z
csc2 HqL z
0
0 - ееееееее
ееее
е
еее
е
2
{
r
0
0
0
The two parameters can be interpreted as the charge ╤ of the body and the
geometric mass m. Of course, in reality, a body of considerable mass has
6. General Relativity
757
no net charge. Therefore, the Reissner?Nordstrom solution is only of
hypothetical interest. However, the Reissner?Nordstrom solution can help
in the study of the more complicated Kerr solution for a rotating black hole
due to the similarity of its structure.
The determinant for the Reissner?Nordstrom solution is the same as for
the Schwarzschild solution. It is plotted in Figure 6.6.8.
detg
=
Simplify[Det[g]]
-r4 sin2 HqL
0
-2
╩g╩ -4
-6
-8
-2
2
2
0 q
-1
0
r
-2
1
2
Figure 6.6.8.
The determinant ╩ g ╩ for the Reissner?Nordstrom solution.
According to the metric of the Maxwell tensor, the energy momentum
tensor reduces to
6.6 Reissner Nordstrom Solution
758
sme = {Q[r] ▒ - O[r] }
8nHrL ь -lHrL<
F = F /. sme
ij
jj
jj
jj
jj
jj
jj
j
k
╤
ее 0 0 yz
- ееее
r2
zz
╤
ееее
ее
0 0 0 zzzz
r2
zz
0
0 0 0 zzzz
z
0
0 0 0{
0
Fc = Fc /.sme
ij 0
jj ╤
jj - ееееее
jjj r2
jj
jjj 0
j
k 0
╤
ееее
ее 0 0 yz
r2
zz
0 0 0 zzzz
zz
0 0 0 zzzz
z
0 0 0{
T = SimplifyA
4 4
1
j
TableA? ? i
j ccccccccc ing@@c, dDD F@@a, cDD F@@b, dDD +
4
S
k
c=1 d=1
1
z
cccccccccccc g@@a, bDD F@@c, dDD Fc@@c, dDDy
z,
16 S
{
8a, 1, 4<, 8b, 1, 4<EE
╤ Hr -2 m r+╤ L
ij ееееееееееееееее
ееееееееееееееееееееее
0
8 p r6
jj
jj
jj
╤2
ееееееееееееее
0
- ееееееееееееееееееееееееееееееее
jj
8 p r2 Hr2 -2 m r+╤2 L
jj
jj
jj
0
0
jj
jj
j
0
0
k
2
2
2
0
0
╤
ееееееее
еееее
8 p r2
2
0
yz
zz
zz
zz
0
zz
zz
zz
zz
0
zz
zz
2
2
╤ sin HqL z
ееееееее
ееееееее
ееее
е
е
2
8pr
{
0
We have so far calculated all quantities sufficient to verify the field
equations in a modified form:
6. General Relativity
759
Simplify[Table[ Ricci[a,b,g,ing] - 8 S
T[[a,b]],{a,1,4},{b,1,4}]]
0
jij
jj 0
jj
jj
jj 0
jj
j
k0
0
0
0
0
0
0
0
0
0
0
0
0
zyz
zz
zz
zz
zz
zz
z
{
The field equations in their original forms are verified as follows:
Simplify[ Table[Einstein[a,b,g,ing] - 8 S
T[[a,b]],{a,1,4},{b,1,4}] ]
ij 0
jj
jj 0
jj
jj 0
jj
jj
k0
0
0
0
0
0
0
0
0
0
0
0
0
yz
zz
zz
zz
zz
zz
zz
{
As a consequence, the Ricci scalar obviously vanishes:
Simplify[RicciScalar[g,ing]]
0
6.7 Exercises
1. Extend the databases in the package PerihelionShift' to other
planets and planetary systems.
2. Find a representation of the perihelion shift using the classical
parameters of an orbit. Compare your calculations to the approximations given in literature.
3. Change the package LightBending' in such a way that you are able
to treat arbitrary masses in the calculations of light bending.Caution:
Save the package before making changes in the program!
760
6.7 Exercises
4. Create a three-dimensional representation of the relation for light
bending (6.53) which considers changes in the mass and diameter of
the star.
5. The line element in a three-dimensional space in a particular coordinate system is
ds2 = dx21 + x1 dx22 + x1 sin2 Hx2 L dx23 .
First, identify the coordinates and, second, examine the flatness of the
metric.
6. The Minkowski line element in Minkowski coordinates
xa = Hx0 , x1 , x2 , x3 L = Ht, x, y, zL
is given by
ds2 = dt2 - dx2 - dy2 - dz2 .
Is the metric flat? Determine the metric tensor.
7. Find the nonzero components of the Christoffel symbols Gab c of
Bondi's radiating metric:
V
ds2 = I еееее
e2 b - U 2 r2 e2 g M du2 + 2 e2 b du dr +
r
2 U r2 e2 g du dq - r2 He2 g dq2 + e-2 g sin2 HqL df2 L,
where V , U , b and g are four arbitrary functions of the three coordinates u, r, and q.
8. Verify that the Kerr form is a solution of the Einstein field equations. The Kerr form is
ds2 = dt2 - dx2 - dy2 - dz2 3
2mr
r
ееееееее
ееееееееее Hdt2 + ееееееее
еееее Hx dx + y dyL +
r4 +a2 z2
a2 +r2
2
a
z
ееееееее
a2 +rееее2е Hy dx - x dyL + еееrе dzL ,
where m and a are constants.
9. Check that the Boyer?Lindquist form of Kerr's solution is a solution
of Einstein's field equations
D
2
ее Hdt - a sin2 HqL dfL ds2 = ееее
r2
2
r
sin HqL
2
2
2
2
2
ееееееее
r2ееееее HHr + a L df - a dtL - ееее
Dее dr - r dq ,
2
2
where r2 = r2 + a2 cos2 HqL and D = r2 - 2 m r + a2 .
6. General Relativity
761
6.8 Packages and Programs
6.8.1 EulerLagrange Equations
This section gives some support in calculating the Euler?Lagrange
equations. First, the notation package is loaded.
<< Utilities`Notation`
Then, the path where you have located the package follows. Please change
the path if you have stored the package in a different directory
$EulerLagrangePath =
$AddOnsDirectory <> "ЙApplicationsЙEulerLagrangeЙ";
AppendTo@$Path, $EulerLagrangePathD;
The next line loads the package.
<< EulerLagrange.m
=================================================
EulerLagrange? 1.0 HDosЙWindows╝L
╘ 19922005 Dr. Gerd Baumann
Runs with Mathematica╝ Version 3.0 or later
Licensed to one machine only, copying prohibited
=================================================
Here, we define a symbolic notation for the function
,
NotationA
x_
u_ @den_D
y EulerLagrange@den_, u_, x_DE
762
6.8 Packages and Programs
The following pallet allows you to generate the shorthand notation for the
Euler?Lagrange operator. You can generate the pallet by selecting the
following cell and use the File+Generate Pallet from Selection button to
activate the pallet.
, @fD
f
f
6.8.2 PerihelionShift
This package calculates the perihelion shift for different planets. The
planets are collected in a database which can be extended by the user.
BeginPackage@"PerihelionShift`"D;
Clear@e1, e2, e3, g2, g3, omega1, omega2,
Orbit, orbit, Energy, AngularMomentum,
PerihelionShift, Planets, D0Orbit, SchwarzschildD;
Planets::usage = "Planets@planet_StringD
creates a list of data for planets and
planetoids stored in the data base of the
package PerihelionShift. The data
base contains the names of the planets,
their major axes, their eccentricity
and the mass of the central planet.
Planets@'List'D creates a list of the
planets in the data base. Planets@'name'D
delivers the data of the planet
given in the argument.";
orbit::usage =
"orbit@phiend_,minorAxes_,majorAxes_,mass_D
creates a graphical representation of
the perihelion shift if the major and
minor axes and the mass are given.";
Orbit::usage = "Orbit@planet_StringD
6. General Relativity
creates a graphical representation of the
perihelion shift for the planets
contained in the data base.";
PerihelionShift::usage =
"PerihelionShift@minorAxes_,majorAxes_,mass_D
Calculates the numerical value
of the perihelion shift.";
AngularMomentum::usage =
"AngularMomentum@minorAxes_,majorAxes_,mass_D
calculates the angular momentum of a planet.";
Energy::usage = "Energy@minorAxes_,majorAxes_,mass_D
calculates the energy of a planet.";
D0Orbit::usage = "D0Orbit@planet_String,
phiend_,options___D plots the orbit
in the case of vanishing determinants Hsee textL.";
Begin@"`Private`"D;
Hdata bases of several planetsL
data =
88"Mercury", 0.5791 10 ^ H11L, 0.2056, MassOfTheSun<,
8"Venus", 1.0821 10 ^ H11L, 0.0068, MassOfTheSun<,
8"Earth", 1.4967 10 ^ H11L, 0.0167, MassOfTheSun<,
8"Icarus", 1.61 10 ^ H11L, 0.827, MassOfTheSun<,
8"Mars", 2.2279 10 ^ H11L, 0.093, MassOfTheSun<,
8"Ceres", 4.136 10 ^ H11L, 0.076, MassOfTheSun<,
8"Jupiter", 7.78 10 ^ H11L, 0.048, MassOfTheSun<,
8"Saturn", 14.27 10 ^ H11L, 0.056, MassOfTheSun<,
8"Uranus", 28.70 10 ^ H11L, 0.047, MassOfTheSun<,
8"Neptune", 44.96 10 ^ H11L, 0.009, MassOfTheSun<,
8"Pluto", 59.10 10 ^ H11L, 0.25, MassOfTheSun<,
8"PSR1916", 7.0204020286 10 ^ H8L, 0.6171313,
2.82837 MassOfTheSun<, 8"TestPlanet",
5.2327 10 ^ H8L, 0.6171313, 2828.37 MassOfTheSun<<;
Hinformation on the planetsL
763
764
6.8 Packages and Programs
Planets@planet_StringD :=
Block@8gh, kh, ma<, MassOfTheSun = 1.993 10 ^ H30L;
If@planet m "List",
Print@DisplayForm@GridBox@Prepend@
Map@Map@PaddedForm@#, 85, 3<D &, #D &, dataD,
8StyleForm@"planet", FontWeight ▒ "Bold"D,
StyleForm@"mean radius",
FontWeight ▒ "Bold"D, StyleForm@
"eccentricity", FontWeight ▒ "Bold"D,
StyleForm@"mass", FontWeight ▒ "Bold"D<D,
RowLines ▒ True, ColumnLines ▒ True,
GridFrame ▒ True,
ColumnAlignments ▒ 8Left<DDD, gh = 0;
kh = 0;
ma = 0;
Do@If@planet m data@@k, 1DD, Planet = data@@k, 1DD;
gh = data@@k, 2DD;
kh = N@data@@k, 2DD Sqrt@1 data@@k, 3DDDD;
ma = data@@k, 4DD;
Print@DisplayForm@
GridBox@88data@@k, 1DD, " "<, 8"mass", ma<,
8"minor axes", kh<, 8"major axes", gh<,
8"eccentricity", data@@k, 3DD<<, RowLines ▒
True, ColumnLines ▒ True, GridFrame ▒ True,
ColumnAlignments ▒ 8Left<DDD, gh = gh;
kh = kh;
ma = maD, 8k, 1, Length@dataD<D;
MajorAxes = gh;
MinorAxes = kh;
Mass = ma;
If@gh ° 0, PerihelionShift@kh, gh, maD, 0DDD;
HSchwarzschild radiusL
SchwarzSchild@mass_D :=
Block@8Gravitation, SpeedOfLight<,
Gravitation = 6.6732 10 ^ H11L;
SpeedOfLight = 2.9979250 10 ^ 8;
2 Gravitation mass Й SpeedOfLight^ 2D;
Hroots of the characteristic polynomialL
6. General Relativity
765
e2@minorAxes_, majorAxes_, mass_D :=
Block@8Schwarzschild, eh<,
Schwarzschild = SchwarzSchild@massD;
eh = H1 3 majorAxes Schwarzschild Й minorAxes^ 2
H1 Sqrt@majorAxes^ 2 minorAxes^ 2D Й
majorAxesLL Й 12D;
e3@minorAxes_, majorAxes_, mass_D :=
Block@8Schwarzschild, eh<,
Schwarzschild = SchwarzSchild@massD;
eh = H1 3 majorAxes Schwarzschild Й minorAxes^ 2
H1 + Sqrt@majorAxes^ 2 minorAxes^ 2D Й
majorAxesLL Й 12D;
e1@minorAxes_, majorAxes_, mass_D :=
Block@8<, He3@minorAxes, majorAxes, massD +
e2@minorAxes, majorAxes, massDLD;
Hg2 and g3 of the Weierstrass functionL
g2@minorAxes_, majorAxes_, mass_D :=
Block@8<, 2 He1@minorAxes, majorAxes, massD ^ 2 +
e2@minorAxes, majorAxes, massD ^ 2 +
e3@minorAxes, majorAxes, massD ^ 2LD;
g3@minorAxes_, majorAxes_, mass_D :=
Block@8<, 4 e1@minorAxes, majorAxes, massD e2@minorAxes, majorAxes, massD e3@minorAxes, majorAxes, massDD;
Hfrequencies of the Weierstrass functionL
omega1@minorAxes_, majorAxes_, mass_D :=
Block@8integrand, x, om1, e11, e21, e31, module<,
integrand = 4 x ^ 3 g2@minorAxes, majorAxes, massD x g3@minorAxes, majorAxes, massD;
integrand = 1 Й Sqrt@integrandD;
e11 = e1@minorAxes, majorAxes, massD;
e21 = e2@minorAxes, majorAxes, massD;
e31 = e3@minorAxes, majorAxes, massD;
module = He31 e21L Й He11 e21L;
766
6.8 Packages and Programs
om1 = EllipticK@moduleD Й Sqrt@e11 e21DD;
omega2@minorAxes_, majorAxes_, mass_D :=
Block@8integrand, x, om2, e11,
e21, e31, module<, integrand =
Abs@4 x ^ 3 g2@minorAxes, majorAxes, massD x g3@minorAxes, majorAxes, massDD;
integrand = 1 Й Sqrt@integrandD;
e11 = e1@minorAxes, majorAxes, massD;
e21 = e2@minorAxes, majorAxes, massD;
e31 = e3@minorAxes, majorAxes, massD;
module = He31 e21L Й He11 e21L;
module = 1 module;
om2 = I EllipticK@moduleD Й Sqrt@e11 e21DD;
Hcreates the orbit
from the orbit parametersL
orbit@phiend_, minorAxes_, majorAxes_, mass_, planet_D :=
Block@8Schwarzschild, bh, omega3, l2, l3, l4, l5,
phi<, Schwarzschild = SchwarzSchild@massD;
om1 = omega1@minorAxes, majorAxes, massD;
om2 = omega2@minorAxes, majorAxes, massD;
omega3 = om1 + om2;
l2 = g2@minorAxes, majorAxes, massD;
l3 = g3@minorAxes, majorAxes, massD;
l4 = Chop@WeierstrassP@phi omega3, 8l2, l3<DD;
l5 = 1 + 12 l4;
bh = Re@3 Schwarzschild Й l5D;
ParametricPlot@8Cos@phiD bh, Sin@phiD bh<, 8phi, 0,
phiend<, PlotRange ▒ All, AspectRatio ▒ Automatic,
Prolog ▒ Thickness@0.001D, PlotLabel ▒ planetDD;
Hcreates the orbit with the data baseL
Orbit@planet_StringD := Block@8<, Planets@planetD;
orbit@6 Pi, MinorAxes, MajorAxes, Mass, planetDD;
Hnumerical value of the perihelion shiftL
PerihelionShift@minorAxes_, majorAxes_, mass_D :=
Block@8ph, ph1<, ph =
6. General Relativity
767
N@2 Homega1@minorAxes, majorAxes, massD PiL, 16D;
ph1 = ph 2.06264806245 10 ^ 5;
Print@" "D;
Print@" Perihelion shift = ", ph1, " arcs"D;
phD;
Hconstants of motionL
AngularMomentum@minorAxes_, majorAxes_, mass_D :=
Block@8Schwarzschild, ll<,
Schwarzschild = SchwarzSchild@massD;
ll = g2@minorAxes, majorAxes, massD;
ll = Schwarzschild Й H2 H1 Й 12 llLLD;
Energy@minorAxes_, majorAxes_, mass_D :=
Block@8Schwarzschild, energy, l2, l3<,
Schwarzschild = SchwarzSchild@massD;
l2 = g2@minorAxes, majorAxes, massD;
l3 = g3@minorAxes, majorAxes, massD;
energy = 2 Sqrt@H1 Й 54 l2 Й 6 l3L Й H1 Й 12 l2LD Й
SpeedOfLightD;
Hasymptitic orbitsL
D0Orbit@planet_String, phiend_, options___D := Block@
8Schwarzschild, e0, n2, phi<, Planets@planetD;
Schwarzschild = SchwarzSchild@MassD;
e0 = 1 Й 24 Schwarzschild Й H4 MajorAxesL;
n2 = 3 e0;
bh1 = 4 Й Schwarzschild
H1 Й 12 + n2 Й 3 n2 Й Cosh@Sqrt@n2D phiD ^ 2L;
bh1 = 1 Й bh1;
ParametricPlot@8Cos@phiD bh1, Sin@phiD bh1<,
8phi, phiend, phiend<, optionsDD;
End@D;
EndPackage@D;
6.8.3 LightBending
768
6.8 Packages and Programs
This package determines the bending of a light beam in a gravitational
field.
BeginPackage["LightBending`"];
Remove[e1, e2, e3, g2, g3, omega1, omega2, Orbit,
Deviation];
Deviation::usage = "Deviation[radius_,mass_]
calculates the numerical value
of the light bending in a gravitational field of a
planet with mass M in a
distance radius of the center.";
Orbit::usage = "Orbit[radius_,mass_] plots the orbit
of a light beam near
a mass in the distance radius. The calculation is
done in Schwarzschild
metric.";
MassOfTheSun::usage;
RadiusOfTheSun::usage;
Begin["`Private`"];
(* --- mass and radius of the sun --- *)
MassOfTheSun = 1.993 10^(30);
RadiusOfTheSun = 7 10^8;
(* --- Schwarzschild radius --- *)
SchwarzSchild[mass_]:=
Block[{Gravitation,SpeedOfLight},
Gravitation = 6.6732 10^(-11);
SpeedOfLight = 2.9979250 10^8;
Schwarzschild = 2 Gravitation
mass/SpeedOfLight^2
];
(* --- roots of the characteristic polynomial --- *)
e1[radius_,mass_]:=
Block[{eh,e31},
e21 = e2[radius,mass];
6. General Relativity
769
eh = N[-1/2 e21 + Sqrt[3] Sqrt[1-36
e21^2]/12]];
e2[radius_,mass_]:=
Block[{Schwarzschild,eh},
Schwarzschild = SchwarzSchild[mass];
eh = -1/12 (1 - 3 Schwarzschild/radius)
];
e3[radius_,mass_]:=
Block[{eh},
eh = N[-(e2[radius,mass] + e1[radius,mass])]];
(* --- frequencies of the Weierstrass function --- *)
omega1[radius_,mass_]:=
Block[{om1,e11,e21,e31,modulus},
e11 = e1[radius,mass];
e21 = e2[radius,mass];
e31 = e3[radius,mass];
modulus = (e21-e31)/(e11-e31);
om1 = EllipticK[modulus]/Sqrt[e11-e31]
];
omega2[radius_,mass_]:=
Block[{om2,e11,e21,e31,modulus},
e11 = e1[radius,mass];
e21 = e2[radius,mass];
e31 = e3[radius,mass];
modulus = (e21-e31)/(e11-e31);
modulus = 1 - modulus;
om2 = I EllipticK[modulus]/Sqrt[e11-e31]
];
(* --- g2 and g3 of the Weierstrass function --- *)
g2[radius_,mass_]:=Block[{},N[1/12]];
g3[radius_,mass_]:=Block[{},
4 e1[radius,mass] e2[radius,mass]
e3[radius,mass]];
(* --- creates the path of the light beam --- *)
Orbit[radius_,mass_]:=
Block[{Schwarzschild,bh,l2,l3,l4,l5,phi,phia,deltaphi
,
770
6.8 Packages and Programs
erg,omega3},
Schwarzschild = SchwarzSchild[mass];
om1 = omega1[radius,mass];
om2 = omega2[radius,mass];
omega3 = om1 + om2;
l2 = g2[radius,mass];
l3 = g3[radius,mass];
l4 = WeierstrassP[phi-omega3,{l2,l3}]+1/12;
erg = FindRoot[l4==0,{phi,Pi/2}];
phia = phi /. erg;
phia = Re[phia];
l4 = Re[WeierstrassP[phi-omega3,{l2,l3}]];
l5 = 1 + 12 l4;
bh = 3 Schwarzschild/l5;
ParametricPlot[{Cos[phi] bh,Sin[phi] bh},
{phi,-phia 0.9,phia 0.9},
Prolog->Thickness[0.001],Ticks->False]
];
(* --- determination of the deviation angle --- *)
Deviation[radius_,mass_]:=
Block[{Schwarzschild,om1,om2,omega3,l2,l3,l4,phi,
deltaphi,dphi,phia,erg},
Schwarzschild = SchwarzSchild[mass];
om1 = omega1[radius,mass];
om2 = omega2[radius,mass];
omega3 = om1+om2;
l2 = g2[radius,mass];
l3 = g3[radius,mass];
l4 = WeierstrassP[phi-omega3,{l2,l3}]+1/12;
erg =
FindRoot[l4==0,{phi,Pi/2},AccuracyGoal\[Rule]34,Worki
ngPrecision\[Rule]34,
MaxIterations\[Rule]50];
phia = phi /. erg;
phia = Re[phia];
deltaphi = N[2 phia-Pi,16];
(* --- the factor 2.06264806245 10^5 converts radian
to arcsecond --- *)
6. General Relativity
771
dphi = deltaphi
2.06264806245 10^5;
Print[" "];
Print[" Deviation = ",dphi," arcs"];
deltaphi];
End[];
EndPackage[];
7
Fractals
7.1 Introduction
Fractals are, today, a basic tool to phenomenologically describe natural
objects. The properties of these objects can be the length of a border, the
relaxation time spectrum of a process, the geometric structure of trees, the
circumference of cells and so forth. All of the measures derived from such
objects are related to the choice of the scale length with which the object is
examined. Fractals are also a tool to describe natural objects such as
biological and medical objects. Fractals are geometric as well as temporal
objects having a long-lasting history such as the monster curves in
mathematics. Fractals are not only restricted to geometric objects but also
have its appearance in time-dependent processes and differential objects.
The observation by Mandelbrot [7.4] of the existence of a "Geometry of
Nature" has led us to think in a new way about natural objects.
The coastline of Norway, a snowflake in Bavaria, the Mississippi River all
of these share a common characteristic that is very common in nature.
They all have a certain amount of geometric complexity. The boundary of
774
7.1 Introduction
the snowflake is difficult to define in geometric terms. The same holds for
the other objects. Indeed, the snowflake must have a very long perimeter,
but it is a very small geometric structure. The mentioned natural examples
provide, with a little reflection, a crisis of definition. If we define a
geometric measure as the determination of a quantifiable measure of these
examples such as length or area, then the geometric measures of physical
characteristics are hard to establish. In fact, the measure could only be
approached on an operational level; that is if one wants to measure the
length of the perimeter of a snowflake, one would have to know by what
means to measure it. Felix Hausdorff (see Figure 7.1.1) was one of the few
mathematicians who thought about these problems in the 20th century. At
the age of 50, Hausdorff was a well-respected mathematician and well
known as a set theoretician. In 1918, Hausdorff published an important
paper contributing to measure theory. This 22-page article published by
Mathematische Annalen gave a new treatment of Lebesgue measure. He
contributed a large amount of knowledge with his own words "Hierzu
geben wir im folgenden einen kleinen Beitrag". This "little contribution" is
his entire theory of measure and of fractional dimension, presented in a
clear and general form. This article is a gem. Few people have read it, yet
it has brought its author more fame, today, than all the rest of his works put
together. The principal application of his theory concerns a family < of
bounded sets associated with a weight bHU L, where U are the countable
sets; thus bHU L is a function of the diameter rHU L = lHrHU LL, with
lHxL = xa . This functional relation is the point at which Hausdorff defines
his fractional dimension.
Figure 7.1.1.
Felx Hausdorff: born November 8, 1868; died Junuary 26, 1942.
7. Fractals
775
About 60 years after Hausdorff's paper, Benoit Mandelbrot (see Figure
7.1.2) coined the term fractal in his "Geometry of Nature". Mandelbrot
examined a large number of natural, artificial, and geometric objects. He
also introduced numerical experiments to demonstrate the fractal beauty of
mappings. The famous Mandelbrot set is one example demonstrating the
fractal nature by an iterated map. Benoit Mandelbrot is the founding father
of the fractal community incorporating fields from physics, biology,
chemistry, material science, architecture, and so forth. The application of
fractal concepts in today's science is omnipresent in all disciplines.
Figure 7.1.2.
Benoit Mandelbrot: born November 20, 1924.
This chapter introduces the fractal concept for geometric objects. It
discusses the experimental determination of fractal dimensions for
geometric structures. In Section 7.4 a monofractal is generalized to the
notion of multifractals. The renormalization group theory in Section 7.5
makes a link between renormalization and fractality. Section 7.6
introduces a generalization of derivatives to fractional derivatives.
776
7.2 Measuring a Borderline
7.2 Measuring a Borderline
A natural borderline separating two objects can be a complicated curve.
When looking at a distant object governed by a geometrical structure, a
skyscraper, for example, we get the impression that its borderlines are
straight lines. Looking through binoculars, we observe that there are
wrinkles and loops in its borderline, and a closer look reveals that the
object has an even more complicated shape. Following this reasoning, we
may wonder whether natural objects can be described fully by Euclidean
geometry. In fact, nowhere in nature will we observe the idealized straight
line. Nature itself uses straight lines connecting two different points only
as an approximation and on small scales. Objects in our natural
environment have different geometrical structures at different scales of
magnification.
Let us consider a tree as an object of our study. If we are far away from the
tree, we can imagine that the picture we see is similar to a point or a short
line on the horizon. If we get closer to the tree, the appearance changes.
First, we see the extension in a plane, and coming closer, we see the spatial
arrangements of its branches. Up close enough, we recognize small
branches and leaves. The building blocks of a tree are not geometrical
objects like cylinders, balls, cones, and the like. The branches of a tree
exhibit self-similarity: After scaling of a branch, a subbranch forms from
which another subbranch can be scaled, and so on. This type of
self-similar scaling law was discovered by Leonardo da Vinci, who
experimented with this subject back in the 16th century [7.4]. In
modern-day mathematics Benoit Mandelbrot has introduced the term
fractals to describe such scaling laws of self-similarity.
When studying complicated natural objects, we simplify the problem by
considering the three-dimensional object in a projection plane. In the case
of the tree, we study the shadow of the tree in order to reduce the problem.
The picture of the shadow is easily created with Mathematica following
Gray and Glynn [7.1] (see Figure 7.2.3). To construct the tree, simple
building blocks are put together in a self-similar way. The package Tree`
contains all the necessary functions to create branches, branchLine[], to
7. Fractals
777
rotate lines, rotateLine[], and to scale branches, BranchScaling. A listing
of the package is given in the section packages of this Chapter 7. A typical
application of the main function is given below. Here, we generate a tree
consisting of 10 branch generations and a natural coloring of the branches.
Tree@Generation ▒ 10, BranchColor ▒ l1D;
Figure 7.2.3.
Fractal tree.
The result is a tree that you will observe in a similar shape in spring or
autumn.
One of the characteristic properties of a projected tree is the length of its
boundary line. If we choose a fixed yardstick length for determining the
length of the boundary line, we get its total length by the number of
yardsticks multiplied by the length of the yardstick. The mathematical
formula is L= N(╤) ╤, where L is the resulting length, ╤ is the length of the
yardstick, and N(╤) is the number of yardsticks used to cover the boundary.
In a second experiment, we change the length of the yardstick ╤. We again
count a number N(╤) and calculate the length L by the same formula as
above. The first observation we make is that the calculated length L has a
778
7.2 Measuring a Borderline
different value compared to the first measurement. For example, if we
choose the yardstick length measuring our tree to be the vertical height of
the tree, we get a different length compared to measuring the tree with a
small yardstick of about 1 cm. The first measurement of the boundary line
is a very crude estimation of its actual length. The accuracy of the
measurement increases with the decrease in length of the yardstick used.
Not only does the accuracy of the measurement increase, but the numerical
value of the total length L increases as well. The method of measuring the
length of the boundary line by means of a yardstick is called the yardstick
method.
Another method for determining the length of a boundary line is the box
counting method. In this method, the object is superimposed on a lattice
with mesh size ╤. If we count the squares which contain a part of the
boundary and multiply the number of boxes N(╤) by mesh size ╤, we get
an approximated length of the boundary line. Again, we observe that with
decreasing mesh size ╤, the accuracy of the measurement and the total
length L increases. The number of boxes counted in the box counting
method is nearly of the same order as the number of yardsticks in the
yardstick method.
If the length L increases while the yardstick ╤ decreases, the question
arises of whether there exists a finite length of the boundary of the tree. If
the length of the boundary is finite, we expect that the number of
yardsticks N(╤) must increase proportionally to 1 Й ╤ (i.e., N(╤) = LN Й ╤). In
other words, if the length of the boundary is L = NH╤L ╤ = LN , where LN is
a constant for any ╤ь0, we can say that the length is constant. If we apply
this mind game to a natural object and count the number of boxes, we
observe a completely different behavior.
The measurement of natural objects like blood cells or the bronchial tree
using the yardstick or box counting method shows a different relationship
between the yardstick length and the number N(╤). The actual relation
observed in experiments ([7.2, 7.3]) is NH╤L = a ╤-D , where D is a number
greater than 1 for plane objects. If we insert the experimentally observed
relation for the number of yardsticks into the length relation L = N╤, we get
L(╤) = N(╤)╤ = a ╤1-D .
(7.2.1)
7. Fractals
779
This relation applies to any boundary line. For an Euclidean curve which is
smooth and differentiable at any point, we expect that parameter a
represents the finite length LN and that dimension D equals 1 as ╤ь0. For
natural objects, the dimension D is not equal to 1. The property that the
dimension of a natural object is different from its topological dimension
was used by Mandelbrot to define the term "fractal" [7.4]. The
experimental determination of dimension D follows from the slope of a
log-log plot in which the length of the curve is plotted versus the length of
the yardstick. The slope of the plot is equal to 1 - D. In fractal theory, the
quantity
logHLH╤LL
logHaL
еееееееееее - ееееееее
еееееееее
D = 1 + ееееееее
logH1Й╤L
logH1Й╤L
(7.2.2)
is called the fractal dimension. This parameter characterizes the plane
filling of the curve. The tree example used earlier in this chapter is
illustrative for our purposes but too complicated to determine the fractal
dimension by analytical methods. Another example of a fractal object is
the curve as defined by Koch, who at the turn of the century introduced the
mathematical monster known as the Koch snowflake. At the same time,
other mathematicians, including Cantor, Peano, and Weierstrass, discussed
sets of points and curves with very strange properties. An example of the
type of curve is given in Figure 7.2.4, which shows the Koch snowflake.
Using the Koch curve, we can show how the fractal dimension of such a
curve (which is nowhere differentiable) is determined and how
self-similarity occurs. First, we will describe the box counting method used
to determine the fractal dimension. After this experimental approach, we
will return to the more analytic approach for fractal curves.
780
7.2 Measuring a Borderline
Figure 7.2.4.
Koch's snowflake.
7. Fractals
781
7.2.1 Box Counting
As mentioned earlier, the determination of a contour length can be carried
out in different ways. One method to determine the total length of a
contour is the application of the yardstick method to gain an
approximation of the length. Another method which will be elaborated
here in more detail is the box counting method. The box counting method
gained its name from the counting of disjunct boxes or squares in the
plane. The squares or boxes can be replaced by other geometric objects
like spheres, ellipsoids, cylinders, and so forth. The explicit form of the
used basic measuring element is of minor importance in the estimation of
the length of a contour. Here, we use the box counting method to
demonstrate its application to plain objects. We apply the box counting
algorithm in its simplest form to show how the method works and how we
can improve the basic procedure to refine the results.
Box counting is one of the most widely used methods to determine the
fractal dimension. Its popularity is largely due to its relative ease of
mathematical calculation and empirical estimation. The definition goes
back at least to the 1930s and it has been variously termed Kolmogorov
entropy, entropy dimension, capacity dimension, metric dimension,
logarithmic density, and information dimension. We will always refer to
box or box counting dimension to avoid confusion.
Let : be a nonempty bounded subset of 52 and let NH╤L be the number of
sets of diameter at most ╤ which can cover :. We refer to the value as the
box counting dimension or box dimension of : as
logHNH╤LL
ееееееееееее M.
D = lim I ееееееее
logH1Й╤L
╤ь0
(7.2.3)
This version of the definition is widely used empirically. To find the box
dimension of the set :, we can draw a mesh of squares of side ╤ and count
the number NH╤L that overlap the set for various small ╤. The dimension is
the logarithmic rate at which NH╤L increases as ╤ь0 and can be estimated
by the gradient of the graph of logHNH╤LL against logH1 Й ╤L.
782
7.2 Measuring a Borderline
The box counting method is based on the division of a plane into squares
of edge length ╤. The box counting method delivers an estimate of the
length of a contour by counting the number of boxes NH╤L of a given size.
Each box containing at least one point is counted in NH╤L. Starting with the
largest ╤ scale (the maximal extension of the object) the grid length ╤ is
decreased successively. In a log-log plot of NH╤L versus ╤, a scaling range
for self-similar structures is obtained.
To demonstrate how this mathematical definitions works in practice, we
will examine each step of the box counting method starting with the
generation of an object, the generation of the squares for different ╤'s, the
counting of the relevant boxes, and the determination of the scaling
exponent.
First, we start with the generation of the object which we will examine.
Suppose we have to measure the contour length of a human cell. The
planar projection of a human cell is mainly described by a disturbed circle.
We assume that the radial coordinate of a circle of radius 1 is increased by
random numbers in the range H0, 0.2L for the x coordinate and H0, 0.1L for
the y coordinate. The sequence of points is generated by the following
table:
points = Table[{Sin[i] + Random[Real,{0,.2}],
Cos[i] + Random[Real,{0,.1}]}
//N,{i,0,2Pi,.05}];
To generate a contour line from these points, we will link each neighboring
points by straight lines. This is carried out by the following function
generating the contour.
Contour[points_]:=Module[{contour},
contour = {};
Do[
AppendTo[contour,Line[{points[[i]],points[[i+1]]}]
],
{i,1,Length[points]-1}];
AppendTo[contour,Line[{Last[points],First[points]}]];
contour
]
7. Fractals
783
The actual contour is then generated by applying this function to the set of
points:
c1 = Contour[points];
A graphical representation of the artificial cell is given next:
pl1 = Show[Graphics[Polygon[points]],
AspectRatio->Automatic];
In the next step, we need to generate the grids allowing us to count the
occupied squares by the contour. The following function generates a
square of total side length lmax divided into subsquares of length ╤.
784
7.2 Measuring a Borderline
Clear[Grid]
Grid[lmax_,eps_]:=Module[{l1={}},
AppendTo[l1,Table[Line[{{-lmax,y},{lmax,y}}],
{y,-lmax,lmax,2 lmax/eps}]];
AppendTo[l1,Table[Line[{{x,-lmax},{x,lmax}}],
{x,-lmax,lmax,2 lmax/eps}]];
l1
]
Using this function, we can generate an animation showing the principal
situation for the measurement process by decreasing the length ╤:
Do[Show[pl1,Graphics[Grid[1.2,eps]],AspectRatio->Auto
matic,
PlotRange->All],
{eps,2,75,5}]
The next step in the determination of the box dimension is to count all
squares occupied by the contour line of the cell. For this step, we have to
7. Fractals
785
check whether the contour line intersects with a specific square or the
square is empty. The following function scans over the total square and
counts the occupied squares:
Clear[PointSearchG];
PointSearchG[lmax_,eps_,points_]:=Module[{
deltaeps,xgmin,xgmax,ygmin,ygmax,occupied,presentPoly
},
deltaeps = 2 lmax/eps;
xgmin = -lmax;
xgmax = xgmin + deltaeps;
ygmin = -lmax;
ygmax = ygmin + deltaeps;
occupied = {};
Do[
Do[
Do[
If[xgmin <= points[[i,1]] < xgmax &&
ygmin <= points[[i,2]] < ygmax,
AppendTo[occupied,{RGBColor[1,1,0],
Polygon[{{xgmin,ygmin},{xgmax,ygmin},
{xgmax,ygmax},{xgmin,ygmax},
{xgmin,ygmin}}]} ]
],
{i,1,Length[points]}];
presentPoly = {RGBColor[1,0,0],
Polygon[{{xgmin,ygmin},{xgmax,ygmin},
{xgmax,ygmax},{xgmin,ygmax},
{xgmin,ygmin}}]};
Show[Graphics[Grid[1.5,eps]],
Graphics[presentPoly],
Graphics[occupied],
Graphics[c1],AspectRatio->Automatic];
xgmin = xgmin + deltaeps;
xgmax = xgmin + deltaeps,
{jx,1,eps}];
xgmin = -lmax;
xgmax = xgmin + deltaeps;
ygmin = ygmin + deltaeps;
ygmax = ygmin + deltaeps,
{jy,1,eps}];
]
786
7.2 Measuring a Borderline
The application of this function to the cell contour demonstrates the
detection and counting of occupied squares
PointSearchG[1.5,10,points]
The numeric counterpart to this graphical representation is realized in the
following function. This function counts the occupied squares and collects
those squares containing a point of the contour in a list. This list is used to
determine the total number of squares for a certain box length ╤.
7. Fractals
787
Clear[PointSearch];
PointSearch[lmax_,eps_,points_]:=Module[{
deltaeps,xgmin,xgmax,ygmin,ygmax,occupied},
deltaeps = 2 lmax/eps;
xgmin = -lmax;
xgmax = xgmin + deltaeps;
ygmin = -lmax;
ygmax = ygmin + deltaeps;
occupied = {};
(* --- detect the occupied squares --- *)
Do[
Do[
Do[
If[xgmin <= points[[i,1]] < xgmax &&
ygmin <= points[[i,2]] < ygmax,
AppendTo[occupied,{RGBColor[1,1,0],
Polygon[{{xgmin,ygmin},{xgmax,ygmin},
{xgmax,ygmax},{xgmin,ygmax},
{xgmin,ygmin}}]} ];
Return[]
],
{i,1,Length[points]}];
xgmin = xgmin + deltaeps;
xgmax = xgmin + deltaeps,
{jx,1,eps}];
xgmin = -lmax;
xgmax = xgmin + deltaeps;
ygmin = ygmin + deltaeps;
ygmax = ygmin + deltaeps,
{jy,1,eps}];
AppendTo[data,{deltaeps,Length[occupied]}];
occupied
]
To count the squares for decreasing ╤, we iterate this function in a certain
range of ╤. In addition, we graphically represent the measuring process and
the data gained in a sequence of figures.
788
7.2 Measuring a Borderline
dat = {};
data = {};
j = 1;
Do[
Show[
GraphicsArray[{Graphics[{Grid[1.5,n],
PointSearch[1.5,n,points],
c1},AspectRatio->Automatic],
LogLogListPlot[AppendTo[dat,data[[j]]],
PlotStyle->{PointSize[0.02],RGBColor[1,0,0]},
PlotRange->{{0.05,1},{6,130}},
AxesLabel->{"≥","N(≥)"},
DisplayFunction->Identity]}],
AspectRatio->Automatic,DisplayFunction->$DisplayFunct
ion];
j = j + 1,
{n,3,25,5}]
NH╤L
100
70
50
30
20
15
10
╤
0.1
0.15
0.20.30.5
0.71
The result shows that the number of occupied squares increases if ╤ is
decreased. Two remarks of caution are appropriate here. Since the
representation of the cell contour is given by a relatively small number of
points, the accuracy of the gained results are not very high. Second, to
increase the reliability of the estimation, the origin of the grid should be
changed. From the different measurements, a mean value of the occupied
squares can be determined and used in the estimation of the scaling
exponent. To estimate the scaling exponent for the present artificial cell
contour, we can fit the data to a straight line in a log-log representation of
the data.
7. Fractals
789
f1 = Fit[Log[dat],{1,x},x]
2.08955 1.0062 x
The result shows that a small deviation from a straight line occurs. The
scaling law of the artificial cell is shown in the following:
Show[Plot[f1,{x,-3,.5},DisplayFunction->Identity],Lis
tPlot[Log[dat],
PlotStyle->{PointSize[0.02],RGBColor[1,0,0]},
DisplayFunction->Identity],
DisplayFunction->$DisplayFunction,AxesLabel->{"log(1/
≥)","log(N(≥))"}];
logHNH╤LL
5
4.5
4
3.5
3
2.5
-3 -2.5 -2 -1.5 -1 -0.5
1.5
0.5
logH1Й╤L
It is obvious that the gained data can be represented as a straight line in a
log-log plot. However, we observe that a scattering of the data points
around the line occurs. This chitter has two main causes. First, the small
number of data points used in the representation of the cell contour results
in fluctuations of the number of occupied squares. Second, there are two
limits of the scaling region for small and large values of ╤, where a major
deviation from the straight line occurs. In the range of large ╤ we have a
cutoff at the diameter of the cell where the scaling relation fails. For very
small ╤, we reach a region where the discrete representation of the contour
cannot be resolved by the box length due to lack of points. Thus, only in
790
7.2 Measuring a Borderline
the middle where the box length and the number of points of the contour
are commensurable, the scaling behavior is observed. The lower and upper
limits in ╤ are thus determined by the extension of the object itself and the
resolution of the contour discretization. The experimental determination of
fractal dimensions by means of the box counting method should only be
trusted if a range of two or three decades in the box length is spanned.
7.3 The Koch Curve
We have been discussing self-similarity, especially of self-similar curves,
but have not explained what is meant by a self-similar object. An example
of a self-similar object from geometry is the congruent triangle. Everybody
knows that the theorem by Pythagoras, c2 = a2 + b2 , is satisfied for a right
triangle. In this formula, c denotes the hypotenuse and a and b represent
the legs of a right triangle (see Figure 7.3.5). The proof of the Pythagorean
theorem is given by the self-similar properties of the triangle.
The area of a right triangle is determined by the length of the hypotenuse
and the smaller of the two angles between the hypotenuse and its legs f
(i.e., F = f Hc, fL). Since F has the dimension of area and c has the
dimension of length, we can write F = c2 F(f). Drawing the normal line of
the hypotenuse through the right angle, we divide the total triangle into
two self-similar triangles (see Figure 7.3.5). The areas of the self-similar
triangles are F1 = a2 F(f) and F= b2 F(f), where F(f) is the same function
for both (similar) triangles. The sum of the areas F1 and F2 is the total area
F of the triangle:
7. Fractals
791
f
a
b
f
f
c
Figure 7.3.5.
Self-similarity on a rectangular triangle.
F = F1 + F2 ,
c2 FHfL = a2 F(f) + b2 F(f).
(7.3.4)
(7.3.5)
Cancellation of the mutual function yields
c2 = a2 + b2
(7.3.6)
QED.
This sort of self-similarity is known as congruence in geometry. If we
apply this construction again to divide the right triangle for each triangle
and repeat the procedure ad infinitum, we get a sequence of triangles
which are scaled versions of the original triangle. At each level of division,
we find the same triangles, but scaled by a different factor. This behavior
of repetition and scaling was used by Helge von Koch to construct the
Koch curve.
The initial element of the Koch curve is a straight line of length LN =1. The
first step in constructing the Koch curve is a scaling of the total length by a
factor r = 1/3. In the second step, four elements are arranged as shown in
Figure 7.3.6. From this figure, we see that the curve loses its
differentiability at the connection points of the four lines. These two
fundamental steps can be infinitely applied to each of the line elements. In
a kth iteration step, we get a total scaling factor of rk = H1 Й 3Lk . The
number of line elements increases up to Nk = 4k . The first three steps of
this construction are shown in Figures 7.3.6-7.3.8. If we measure the
792
7.3 Koch Curve
length of the Koch curve by a yardstick of the same length as the scaling
factor ╤ = r, we find the equation from the length relation LH╤L = N(╤) ╤,
logHNL
еееееееее
D = ееееееее
logH1Й╤L
(7.3.7)
and one obtains D = logH4L Й logH3L = 1.218 .... for the Koch curve. Thus,
the fractal dimension for a self-similar curve follows from the number of
building blocks N of the generator and the scaling factor r, which is used
as the yardstick length. The geometrical structure of the line elements is
not contained in the fractal dimension because the fractal dimension is not
a unique property of a curve. Thus, we get the same fractal dimension for
curves with completely different appearances (compare Figures 7.3.8 and
7.3.9).
The Koch curves of the Figures 7.3.6-7.6.9 are constructed in
Mathematica with the function Line[]. We define the generator of the
Koch curve in the Koch[] function, which is part of the Koch` package,
and use the Mathematica function Map[] to generate the higher iterations
of the generator (see Section 7.8.2). By keeping the generator and the
iteration separate in the creation process of the fractal curve, we are able to
mix two or more generators into the iteration process. In Figure 7.3.10 the
Koch generator is mixed with a rectangular representation. The first two
iterations are done with the original Koch generator. The next two
iterations use the rectangular Koch generator. Separating the iteration
process from the definition of the fundamental generators allows any
mixing of generators in any state of the iteration. In package Koch`, we
define a number of generators of fractal curves. Their combinations are
accessed by the function Fractal[]. This function uses a string containing
one of the possible fractals as the first argument. The second argument of
the function changes the default values of the generators.
Another form of the Koch curve is obtained if we change the base angle a
of the triangle in the generator. If we again use four line elements to set up
the generator and alter the scaling factor to r = 1 Й H2 + 2 cos aL, we find a
fractal dimension of
log 4
еееееееееееееееееееееее .
D = ееееееееееееееее
log 2+logH1+cos aL
(7.3.8)
A representation of the dimension D versus the angle a is given in Figure
7.3.11. In the case of a = 0, the dimension is reduced to D = 1 and for
7. Fractals
793
a = p Й 2, the maximum dimension D = 2 occurs. For D = 2, we have a
plane filling curve. For the specific value a = 1.4, the sixth iteration of the
Koch curve with a variable base angle is given in Figure 7.3.12.
Figure 7.3.6.
First iteration of the Koch curve.
Figure 7.3.7.
Second iteration of the Koch curve.
Figure 7.3.8.
Third iteration of the Koch curve.
794
7.3 Koch Curve
Figure 7.3.9.
Fourth iteration of an altered Koch curve. The triangle is located at the right end of the unit
base element.
Fractal@"Mixture"D
Figure 7.3.10.
Mixing of two generators. The first two iteration steps are governed by the original Koch
generator. In the last two iteration steps, a rectangular Koch generator is used.
Ds
2
1.8
1.6
1.4
1.2
0.25 0.5 0.75
Figure 7.3.11.
1
1.25 1.5
Change of the fractal dimension under a change of base angle.
a
7. Fractals
795
Fractal@"WKoch", Angle > 1.4, Generations > 5D
Figure 7.3.12.
Koch curve with base angle a =1.4. The scaling factor is r = 0.42736....
7.4 Multifractals
In the previous sections, we discussed structures with mutual scaling
factors. This kind of self-similarity is a special case of fractals. A more
common type of fractal uses several scaling factors in competition with
one another. If in the same system different scaling factors occur with
different probabilities, we speak of multifractal behavior. The first step in
the construction of a multifractal consists of the division of a set into j
components, in which each is scaled by the factor 1 Й r j < 1. We assume
that each part of the j-fold set is related to a probability P j . The
probabilities P j are normalized so that ?nj=1 P j = 1, where n counts the
number of subsets. The second step in constructing k = 2 is a repetition of
the first step applied to each subset. The n subsets are each divided into n
subsets and are related to the corresponding probabilities. A graphical
representation of this division is given in Figure 7.4.13. The multifractal is
created as kь╤.
796
7.4 Multifractals
Figure 7.4.13.
Representation of a multifractal. The initial state k = 0 and the first iteration k = 1 are
shown. The scaling factors are r1 , r2 , and r3 . The related probabilities are P1 , P2 , and P3 .
The consequence of this construction is that we can divide the total fractal
into n parts. Each part of the fractal is scaled by a factor 1 Й r j and the
measure of the jth part is determined by P j . Using these quantities, we can
define one of the characteristic functions of a multifractal by
q
q
N
p j,i H╤L = P j cq H╤r j L,
cq, j H╤L=?i=1
(7.4.9)
where cq, j (╤) characterizes the jth part of the fractal by a probability
p j,i (╤) ( p j,i H╤L is the ith probability for the jth part of the total fractal). For
the total fractal, we get
cq H╤L=?nj=1 cq, j H╤L.
(7.4.10)
Using the relation cq H╤L = ╤Hq-1L Dq and Eq. (7.4.9), we get the expressions
q Hq-1L Dq
q
cq, j H╤L =P j cq H╤r j L =P j r j
cq = ?
n
j=1
q Hq-1L Dq
Pj rj
╤Hq-1L Dq ,
╤Hq-1L Dq ,
(7.4.11)
(7.4.12)
which define the implicit equation for determining the generalized
dimension Dq by
n
q Hq-1L Dq
? j=1 P j r j
= 1.
(7.4.13)
Depending on the choice of probabilities P j and scaling factors r j , we can
use Eq. (7.4.13) to derive several special cases for a multifractal. For q = 0
we get the fractal dimension D = D0 . This dimension was introduced by
Mandelbrot [7.4] for a fractal
n
-D
? j=1 r j 0 = 1.
(7.4.14)
7. Fractals
797
For arbitrary q and identical scaling factors r j = r, we get the
representation of Dq by
n
q Hq-1L Dq
? j=1 P j r j
= 1,
q
n
Hq - 1L Dq ln r = -ln ? j=1 P j ,
q
n
1 ln ? j=1 P j
Dq = ееее
е
еее
е
ееееееееееееееее
еееее
.
1
q-1
ln ееееr
(7.4.15)
(7.4.16)
(7.4.17)
Once the probabilities P j and the scaling factors r j are equal for each
individual j, the multifractal properties no longer occur.
Knowing the dependence of Dq on q, alternate representations of the
fractal dimensions emerge. By a Legendre transformation, we can
introduce
Hq - 1L Dq = HtHqLL = q aq - fq ,
(7.4.18)
where fq is the multifractal distribution and aq is the HЖlder exponent. The
HЖlder exponent aq is defined by the derivative of tHqL:
d
ееее tHqL.
aq = ееее
dq
(7.4.19)
Once we know the fractal dimensions Dq , we are able to determine the
HЖlder exponent and fq by relations (7.4.19) and (7.4.18), respectively.
Knowing both quantities, we can plot f = f HaL versus a, eliminating q.
Calculating the derivative of tHqL given in Eq. (7.4.19) causes numerical
problems. Finding the numerical derivative of the Legendre transformation
of Dq is the main problem in our calculation. In the package
MultiFractal` (see Section 7.8.3), we use a symmetric difference
procedure (see Section 3.5 of Chapter 3) for representing the numerical
values of the derivatives of tHqL. The transformation to t is defined in the
function Tau[]. The approximations of derivatives by their differences
result in a numerical error, but it is sufficiently small if we choose steps dq
in q as a small quantity.
MultiFractal[] calculates the multifractal characteristics. Probabilities P j
and scaling factors r j are input parameters for this function. The fractal
dimension Dq , the function tHqL, and the Legendre transfor- mation are
determined by the functions Dq[], Tau[], and Alpha[], respectively. After
their calculation, these quantities are graphically represented by the
798
7.4 Multifractals
Mathematica function ListPlot[]. An example of a transformation is given
in Figures 7.4.14-7.4.17.
7.4.1 Multifractals with Common Scaling Factor
We now consider a multifractal with a fixed and a mutual scaling factor
ri = r. To determine the generalized dimensions Dq , we use Eq. (7.4.17),
which gives
1
еееее
Dq = ееее
q-1
q
ln ?n P
j=1 j
ееееееееееееееее
еееее .
ln ееее1
(7.4.20)
r
In the following, we consider a model that contains three independent sets,
n = 3, characterized by the probabilities P1 = 1 Й 5, P2 = 3 P1 , and P3 = P1 .
If we use relation (7.4.17) for these three processes, we get
q
1
еееее
Dq = ееее
q-1
q
q
lnHP1 + P2 +P3 L
ееееееееееееееее
ееееееееееееееее .
lnH1ЙrL
(7.4.21)
Normalizing the probability by P3 = 1 - P1 - P2 simplifies expression
(7.4.21) to
q
1
еееее
Dq = ееее
q-1
q
lnHP + P +H1-P -P Lq L
1
2
1
2
ееееееееееееееееееееееееееееееее
lnH1ЙrLееееееееееееееееее .
(7.4.22)
The numerical results are represented in Figure 7.4.15 which is created by
MultiFractal[{1/5,3/5,1/5},{1/2,1/2,1/2}] . In the above case, probabilities
P1 = P3 and P2 = 3 P1 simplify Eq. (7.4.22) to
1
Dq = ееее
еееее
q-1
lnH2+ 3 L+q lnHP1 L
ееееееееееееееее
ееееееееееееееееееееее .
lnH1ЙrL
q
(7.4.23)
From relation (7.4.23), we can derive analytic relations for the HЖlder
exponent aq , and for the spectrum fq by using relations (7.4.19) and
(7.4.18). We get for aq the expression
1
3q ln 3
ееееее I ееееееее
ееееее - lnHP1 LM.
aq = ееееееее
lnH1ЙrL
2+3q
(7.4.24)
The spectrum of the fractal dimensions is given by
1
d
ееееее Iq ееее
ееее lnH2 + 3q L - lnH2 + 3q LM.
fq = ееееееее
lnH1ЙrL
dq
(7.4.25)
Relation (7.4.25) is independent of P1 and only contains the ratios of the
probabilities. Since the expressions for Dq , aq , and fq can not be solved
explicitly, we use the numerical method implemented in the function
7. Fractals
799
MultiFractal[] to find the solution. Figures 7.4.14?7.4.17 show the results
of our calculation. The fractal dimension D0 of our model is D0 = 1.58... .
Figure 7.4.14 represents the auxiliary function tHqL =Hq - 1L Dq , which is
the basis of the numerical calculations. Figure 7.4.15 contains the
representation of the generalized dimension Dq . Relations (7.4.19) and
(7.4.18) for fq and aq are shown in Figure 7.4.16. We observe that aq is a
monotonically decreasing function and that fq shows its maximum at q =0.
The Legendre transform of these relations results in the function f (a) as
shown in Figure 7.4.17. We observe that the values of f (a) are almost
equally spaced at the maximum and become denser at the boundaries of
the a interval. In the a-╤ limit, the function f (a) tends to 0, but for a╤ , a
finite value f (a) results. This means that for a = a╤ , a finite dimension of
the subsets exists which is smaller than D0 but greater than zero.
t
25
20
15
10
5
-10
5
-5
10
q
-5
Figure 7.4.14.
Function tq = Hq - 1L Dq versus q in the range q ? @-10, 10D for the model fixed by n = 3
and r = 1 Й 2. The probabilities are P1 = 1 Й 5, P2 = 3 Й 5, and P3 = 1 Й 5.
800
7.4 Multifractals
Dq
2.2
2
1.8
1.6
1.4
1.2
-10
Figure 7.4.15.
-5
0.8
5
10
q
Generalized fractal dimension Dq for the model n = 3, r = 1 Й 2, P1 = 1 Й 5, P2 = 3 Й 5,
P3 = 1 Й 5, and q ? @-10, 10D.
a,f
2
1.5
1
0.5
-10
Figure 7.4.16.
-5
5
10
q
The exponent aq (top) and fq (bottom) versus q for the model n = 3, r = 1 Й 2, P1 =P3 = 1/5,
P2 = 3 Й 5, and q ? @-10, 10D.
7. Fractals
801
f
1.5
1.25
1
0.75
0.5
0.25
0.75
Figure 7.4.17.
1.25 1.5 1.75
2
2.25
a
The fractal spectrum f HaL for a multifractal with n = 3, r = 1 Й 2, P1 = P3 = 1 Й 5, and
P2 = 3 Й 5.
7.5 The Renormalization Group
Renormalization group theory is useful for describing physical phenomena
that show the same behavior on different scales. We assume that p is a
quantity measured with a certain accuracy. The same physical quantity is
measured in a second experiment, yielding p' with an accuracy which is
smaller by a factor of 2 than the first measurement. We assume there is a
resolution transformation f2 connecting the two measurements by
p ' = f2 H pL,
(7.5.26)
where subscript 2 denotes the order of resolution. If we decrease the
resolution of the measurement by another factor of 2, we get the relation
p '' = f2 H p 'L = f2 H f2 H pLL = f2 Ъ f2 H pL = f4 H pL.
(7.5.27)
The general representation of our resolution transformation for two
arbitrary resolutions a and b is given by
fa Ъ fb = fab ,
f1 = 1,
(7.5.28)
(7.5.29)
where 1 represents the identity transform. Applying the resolution
transformation to any physical state, a reduced state containing less
information is created. Decreasing the resolution from a state with small
802
7.5 Renormalization Group
resolution is, in general, not possible. In other words, the function f
cannot be inverted in general. A set of functions which is not 1 to1 is
called a semigroup in mathematics. In physics, the transformation reducing
the resolution is called renormalization. (Strictly speaking, f should be
called a semirenormalization group.) By definition, the renormalization
group is closely related to the definition of a fractal.
Since a fractal stays invariant under a scaling transformation, it is evident
that a fractal also stays invariant under a renormalization transformation.
Chronologically both terms ? fractal and renorma- lization ? were
introduced in the 1970s. Both describe the behavior of an object with
changing scales. The difference between the two terms is that a fractal is
based on geometrical properties, whereas renorma- lization considers the
physical properties in a scaling process. However, recent developments in
fractal theory also consider physical properties, whereas renormalization
theory is also applied to geometric objects. Consequently, the distinction
between a fractal and renormalization theory is disappearing.
Renormalization theory is a tool describing critical phenomena like phase
transitions in a liquid. Liquids, for example, possess a critical point in their
phase diagrams. Renormalization theory is used to describe the behavior of
the system in the immediate neighborhood of the critical point. Let us
consider a state of liquid below the critical point where a mixture of liquid
and gas coexists. Below the critical point, the mixture contains more liquid
than gas. If we "coarse grain" our observation, we get a system which is
dominated by the liquid phase. The combination of cells containing liquid
and gas components produce a liquid state under renormalized conditions.
The repetition of the "coarse graining" process results in a global liquid
state. If, on the other hand, the initial state of the phase diagram contains
more gas than liquid, the renormalization results in a gaseous state.
In another example, we consider the renormalization procedure in
connection with percolation theory. Percolation theory is a theory
describing the connections in a network of random links. The theoretical
basis for this theory was created by P.G. de Gennes [7.7], winner of the
1991 Nobel Prize. He applied percolation theory for disordered materials
in polymer science. Percolation phenomena are widespread in nature,
occurring in biological, chemical, and physical systems.
7. Fractals
803
Percolation theory allows the connection of two different boundaries with
a cluster of particles on a lattice. Specifically, let us examine the transport
of electrons through a porous medium which is located between two metal
plates. The transport of the charge is carried by a percolation cluster
connecting both plates. In order to study the transport of electrons, picture
the simulation of a current in a porous medium on a two-dimensional
lattice. Atoms carry the charge on the lattice. The atoms are randomly
scattered. Using the probability p, an atom at a certain location on the
lattice can be located.
The renormalization step on this lattice is defined by the rule valid for a 2
в 2 sublattice, which is called the virtual lattice. We are able to replace the
region of the virtual lattice with a new lattice point in the renormalized
lattice. The resultant lattice is called the superlattice. The (2 в 2) cells of
the virtual lattice are called blocks (see Figure 7.5.18).
Figure 7.5.18.
Renormalization steps with (2в2) blocks.
The transition from the original lattice to the superlattice follows rules for
replacing old atoms with new ones. The simplest rule applies if we have
four atoms in a block. In this case, the new point in the super lattice is an
atom. If we only have three atoms in a block, another new atom emerges
on the superlattice. Accordingly, percolation clusters can form horizontally
as well as vertically. If a block only contains one or two particles, it is
impossible for a percolation cluster to occur which is independent of any
direction. Therefore, no atom appears on the superlattice. Applying the
transition rules as defined in a probability projection, we can write down
the probability of finding an atom on the superlattice by
804
7.5 Renormalization Group
H p 'L = f2 H pL = p4 + 4 p3 H1 - pL.
(7.5.30)
The first term describes the probability that all four atoms are present in a
block. The second term takes into account the four possible arrangements
of three atoms in a block. Since we now know the function f2 , we can
determine the phase transition by using the properties of f2 .
Generalizing relation (7.5.30) for a lattice with n = b в b locations on
which m empty points exist is given by the expression
m
f HH pLnm L =?
i=0
i
ij n yz Hn-iL
H1 - pL .
j zp
i
k {
(7.5.31)
Equation (7.5.31) specifies the probability on a lattice if the block contains
n locations of which all m points are empty.
The critical point pc on the (2в2) lattice is defined in such a way that the
probability will not change under the transformation f2 . The fixed point pc
is derived from the relation
pc = p4c + 4 p3c H1 - pc L
(7.5.32)
with solutions
pc = 90, 1,
Х!!!!!!!
1■ 13
ееееееее
еееееееее =.
6
(7.5.33)
The numerical values of the third and fourth solutions are -0.434 and
0.768. Since p is a probability which is always greater than 0, we have to
exclude the solution pc = - 0.434 from the physical solution set. The cases
pc = 0 and pc = 1 are trivial since they correspond to an empty or
occupied lattice. The remaining value of pc = 0.768 seems to be the
critical value for which a percolation takes place. We observe a gap if we
compare the theoretical value with the value pc = 0.59 yielded by
computer simulations. However, the experimentally determined value of
pc = 0.752 is fairly close to its theoretical counterpart [7.5, 7.6]. A
graphical representation of the critical probability versus the number of
lattice points is given in Figure 7.5.19. The curves in this figure represent
different superlattices.
7. Fractals
805
pc
1
0.8
0.6
0.4
0.2
5
Figure 7.5.19.
10
15
20
25
30
m
Percolation probability for super lattices with 4, 8, 16, and 32 lattice points. The
probability is plotted versus the number of empty lattice points.
To see how other solutions of (7.5.33) are reached, we first consider the
case p < pc . In this case, we get the inequalities
pc > p > f2 H pL > f22 H pL > Ъ Ъ Ъ > f2n H pL.
(7.5.34)
Relation (7.5.34) shows that the probability p decreases in each
renormalization step. After infinitely many renormalization steps, we get
the limit f╤ H pL = 0. In other words, a point with an atom somewhere on the
lattice is impossible, since the lattice is empty.
For the case p > pc , the reverse occurs and f╤ H pL = 1. After infinitely
many renormalization steps, the superlattice is fully occupied. This means
that all initial values in the neighborhood of pc = 0.768 will tend to be
pc = 0 or pc = 1. The fixed point at pc = 0.768 is unstable (see Figure
7.5.20).
0
Figure 7.5.20.
pc
1
Stability of the fixed points in the renormalization procedure.
In the following, we determine the fractal dimension of the cluster at
percolation pc = 0.768. If an atom is present on the superlattice, we know
806
7.5 Renormalization Group
that there are either three or four atoms in a block. The expectation value
pc Nc of occupied lattice points is thus given by
H pc Nc L = 4 p4c + 3 Ъ 4 p3c H1 - pc L
С Nc = 4 p3c + 3 Ъ 4 p2c H1 - pc L,
(7.5.35)
(7.5.36)
where Nc is the mean value of atoms provided that the superlattice is
occupied. The general formula for a square grid has the representation
m
Nc HH pLnm L =?
i=0
i
ij n yz
j z Hn - iL pHn-i-1L H1 - pL .
ki{
(7.5.37)
Equation (7.5.37) counts the mean number of occupied lattice points for a
square lattice with n locations and with m empty locations. A graphical
representation of Nc versus m is given in Figure 7.5.21. The curves in the
figure represent different block sizes.
Nc
30
25
20
15
10
5
5
Figure 7.5.21.
10
15
20
25
30
m
Mean number of occupied locations in a square lattice. The block size is 4, 8, 16, and 32 as
shown in the curves from bottom to top.
The meshsize in the superlattice is twice that of the original lattice. If we
divide the meshsize by 2 in the superlattice, we observe Nc atoms, the
average in the original lattice. Generalizing this observation when reducing
the observation scale by a factor of 1 Й b yields
Nc HbL = b-D .
(7.5.38)
In the example discussed above, b = 2. From relation (7.5.38) we get for
the specific case,
7. Fractals
807
ln N
D = ееееееее
еcеее = 1.79,
ln 2
(7.5.39)
where the constant D represents the fractal dimension of the percolation
cluster. D = 1.79 is in good agreement with the value found in computer
simulations. However, the experimental value of the fractal dimension is
different (D = 1.9 [7.5]). Figure 7.5.22 represents the fractal dimension
compared to the empty lattice points for several block sizes. We observe
from this figure that the fractal dimension decreases with an increase of
empty lattice points. The dimension D approaches 2 if the lattice is almost
fully occupied.
D
2
1.75
1.5
1.25
0.75
5
10
15
20
25
30
m
0.5
Figure 7.5.22.
Fractal dimension of a percolation cluster versus empty locations for four block sizes 4, 8,
16, and 32.
In our previous considerations, we calculated the fractal cluster dimension
at the critical point. Other interesting quantities in the neighborhood of the
critical point are the critical exponents. The critical exponents are easy to
derive if we again use the renormalization procedure. As an example, we
determine the critical exponent of the correlation length.
For p < pc and p in the neighborhood of pc , we can represent the
correlation length x by
x = x0 ╩ pc - p ╩-n ,
(7.5.40)
808
7.5 Renormalization Group
where x0 is a characteristic length of the system (e.g., the meshsize). If we
consider the rescaled superlattice, we find for the invariant correlation
length,
x = x0 ё ╩ pc - p' ╩-n
(7.5.41)
with x0 ' = 2 x0 . From Eq. (7.5.39) and (7.5.40), we derive the critical
exponent n:
logH2L
n = ееееееееееееееее
ееееееееееееееееееееее .
logH p L- p'ЙH p - pL
c
c
(7.5.42)
At the limit where p and p' tend to pc , we can replace
pc - p'
≥ p'
ееееееее
еееее Ж ееее
ееее ? .
pc - p
≥p p= pc
(7.5.43)
The final result for the critical exponent is
n=
logH2L
ееееееееееееееее
еееееееееееееееее .
≥ f2 HpL
log I ееееееее
≥ pеееееее M? p= pc
(7.5.44)
Using the functional relation f2 in Eq. (7.5.44), the numerical value n =
1.4 is close to the experimental value of n =1.35.
The renormalization group theory is useful for determining fractal and
critical properties of a system. Note that the renormalization theory is a
kind of perturbation theory. Errors occur in the renormalization procedure
when defining renormalization rules. For example, blocks containing more
than two atoms are replaced by atoms on the superlattice, whereas blocks
containing one or two atoms are given by a void. This coarse graining
process is the source of renormalization errors; that is, we create a crude
picture of the original lattice in the superlattice containing links and gaps
on sites where no links were present in the original lattice (see Figure
7.5.23). To minimize errors, we use large block sizes. If we use blocks of
size b, we have b2 lattice points. The number of states in the block is given
2
by 2b and increases rapidly with block size b. From a practical point of
view, b = 4 is the upper limit for which we can calculate the renormalized
function fb .
The package Renormalization` (see Section 7.8.4) contains the functions
Nc[] for determining the mean number of occupied lattice points, Dim[]
for calculating the fractal dimension, and Pcrit[] for calculating the critical
probability of percolation. Function RenormPlot[] allows the graphical
7. Fractals
809
representation of the above functions. Examples of the plots are given in
Figures 7.5.19, 7.5.21 and 7.5.22.
Figure 7.5.23.
Errors in the renormalization of a 2в2 lattice.
7.6 Fractional Calculus
Fractional calculus, contrary to fractal geometry, is an old subject in
mathematics. This kind of calculus is useful to describe phenomenological
models for different chemical and physical processes. Among these
processes are temporal relaxations of polymeric material and diffusion
processes in space and time. Fractional calculus is an approach to
mathematically describe natural phenomena which are mainly connected to
power law behavior in the limit of large arguments. The power-law
behavior of large arguments for natural systems is typically accompanied
by a deviation from these power laws for small arguments. Thus, fractional
calculus is a tool to interpolate between these two regimes by means of
fractional differentiations.
810
7.6 Fractional Calculus
7.6.1 Historical Remarks on Fractional Calculus
The term fractional calculus is by no means new. It is a generalization of
the ordinary differentiation by noninteger derivatives. The subject is as old
as the calculus of differentiation and goes back to times when Leibniz (see
Figure 7.6.24), Gauъ, and Newton invented this kind of calculation. In a
letter to L`Hospital in 1695, Leibniz raised the following question:
Figure 7.6.24.
Gottfried Wilhelm von Leibniz: born July 1, 1646; died November 14, 1716.
Can the meaning of derivatives with integral order d n yHxL Й dxn be
generalized to derivatives with nonintegral orders, so that, in general,
n ? &? This question goes back to a query of Bernoulli, who was
interested in the noninteger differentiation of a product. The story goes
that L`Hospital was somewhat curious about that question of Leibniz and
replied by another question. What if n = ее12ее ? Leibniz in a letter dated
September 30, 1695 replied: Il y a de l'apparence qu'on tirera un jour des
consequences bien utiles de ces paradoxes, car il n'y a gueres de
paradoxes sans utilitИ. The translation reads: It will lead to a paradox,
from which one day useful consequences will be drawn. The question
raised by Leibniz for a fractional derivative was an ongoing topic in the
last 300 years. Several mathematicians contributed to this subject over the
years. People like Liouville, Riemann, and Weyl made major contributions
to the theory of fractional calculus.
7. Fractals
811
In fact, a fractional derivative is useful for some types of function. For
example, let us consider the nth derivative of a power xm . We know that
the general expression for the nth derivative is given by
n
m
d x
m!
ееееееее
еееее = ееееееее
ееееееее xm-n .
dxn
Hm-nL!
(7.6.45)
We also know that a factorial is connected with Euler's G function by the
relation n != GHn + 1L. Replacing the factorials in Eq. (7.6.45) by the G
function, we can write
n
m
GHm+1L
d x
ееееееее
ееееее xm-n .
dxnеееее = ееееееееееееееее
GHm-n+1L
(7.6.46)
This representation is equivalent to Eq. (7.6.45); however, it contains the
potential of a generalization. We know that the G function is defined for
continuous arguments over the complex domain. If we now change the
integer value of n to a number q ? &, we are able to generalize the
meaning of an integer differentiation to a noninteger form. We can even
define a complex differentiation. Replacing n by q in Eq. (7.6.46) results
in general in
q
m
GHm+1L
d x
ееееееее
еееее = ееееееееееееееее
ееееее xm-q .
dxq
GHm-q+1L
(7.6.47)
Relation (7.6.47) has a well-defined meanin; however, it is restricted to
powers xm . However, if we try to fractionally differentiate such simple
functions with Mathematica, we end up with the following result:
≥8x,1Й2< x2
D::dvar :
1
Multiple derivative specifier 9x, cccc = does not have the form
2
8variable, n< where n is a nonnegative machine integer.
≥9x, ccc1c = x2
2
812
7.6 Fractional Calculus
This shows us that Mathematica is not capable of dealing with fractional
differentiation orders. The developer of Mathematica, however, designed
the system in such a way that the user can extend the definition of
derivatives. This extension will be our subject in the following. Telling
Mathematica that fractional derivatives of powers are useful mathematical
constructs is realized by the following lines:
Unprotect@DD;
First, unprotect the differentiation and then add a new definition:
D@x_m_. , 8x_, q_<D :=
Gamma@m + 1D
cccccccccccccccc
cccccccccccccccc
ccccccccc xmq Й; Head@qD == Real ╩╩
Gamma@m q + 1D
Head@qD == Rational ╩╩ Head@qD == Complex
Protect the differential operator again:
Protect@DD;
The definition of the fractional derivative of powers is based on Eq.
(7.6.47) and restricts the order of differentiation either to the rational, the
real or the complex numbers. An example for a rational number reads
≥9x, ccc1c = x
2
Х!!!!
2 x
ееееееее
ееееееееее
Х!!!!
p
If we set the order of differentiation q to a real number, we find
≥8x,2.1< x2
1.87156
ееееееееееееееее
ееееееееее
x0.1
7. Fractals
813
Even if we use complex numbers differentiation order, we get a result:
≥8x,11.5+I< x4
H57152.1 - 143371. бL x-7.5-б
This kind of formula was discussed by Lacroix in 1819 [7.8] based on the
work by Euler in 1738 [7.9]. In retrospect, these formulas are the first
analytical answer of Leibniz's question on fractional derivatives. The
answer lied 100 years dormant and needed the work of Euler to get a
preliminary answer. The story on fractional calculus continued with
contributions from Fourier, Abel, Liouville, Riemann, and Weyl. For a
historical survey, the reader can consult the books of Oldham and Spanier
[7.10] or Miller and Ross [7.11]. The historical developments culminated
in two main calculi based on the work of Riemann [7.12] and Liouville
[7.13] on the one hand and on the work of Weyl [7.14] on the other hand.
Both formulations are connected and Weyl's calculus forms a subset of the
Riemann?Liouville (RL) calculus. In Section 7.6.2 we will discuss the RL
calculus. Section 7.6.3 is concerned with the Mellin transform used in the
solution of fractional differential equations. Section 7.6.4 discusses the
solution of different fractional differential equations.
7.6.2 The Riemann?Liouville Calculus
The development of fractional calculus within the framework of classical
functions is well known and no purpose would be served by a detailed
exposition. However, the present subsection has the aim to provide the
reader with the basic tools to carry out such calculations by computer. We
not only present the theoretical background of the calculus but also show
how symbolic computation is instrumental in calculating fractional
expressions. Most of the basic analysis is discussed in the book by Oldham
and Spanier [7.10]. The more theoretical issues as well as historical
remarks are collected in the book by Miller and Ross [7.11].
In the previous subsection, we introduced the fractional derivative by
heuristics using properties of Euler's G function. In this subsection, we will
814
7.6 Fractional Calculus
define an operator to calculate fractional derivatives. This operator is
based on works by Riemann and Liouville (RL). Paradoxically, the basis
of this differential operator is not a derivative but an integral. However, we
can understand an integration as a differentiation if we introduce a
differentiation with negative exponents. For example the negative
first-order derivative is defined by
-1
x
d
ееееееее
ее f HxL := ?0 f HtL ? t.
dx-1
(7.6.48)
The negative second-order derivative is
-2
x t
d
ееееееее
ее f HxL := ?0 ?0 f HsL ? s ? t ?
dx-2
(7.6.49)
The negative order of differentiation means nothing more than an
integration. Higher orders of differentiation are calculated by nesting the
integrals on the right-hand side. We will abbreviate this kind of recursion
by the symbol +-n
0, x , where n is a positive integer. Thus, Eq. (7.6.48) is
reduced to
x
+-1
0, x f HxL = ?0 f HtL ? t.
(7.6.50)
The symbol +-n
0, x contains the complete information for the calculation of
the negative differential in a nutshell. The lower two indices denote the
lower and upper boundaries of the integral. The superscript represents the
order of differentiation. A weak generalization of the above notation is
gained if we allow an arbitrary starting point a as the lower boundary in
the integral; that is,
x
+-1
a, x f HxL = ?a f HtL ? t.
(7.6.51)
If we consider the nth derivative +-n
a, x of an arbitrary function f HxL, we
write
x x
+-n
a, x f HxL = ?a ?a n-1 f Hx0 L ? xn-1 ? ? x0 .
(7.6.52)
Recalling Cauchy's integral formula
n
d
n!
-n-1
ееее
f HzL ? z,
nе f HxL = еееееееее ? Hz - zL
dxееее
2 pi C
(7.6.53)
we can reduce Eq. (7.6.52) to
1
x
+-n
ееееее Hx - x0 Ln-1 f Hx0 L ? x0 .
a, x f HxL = ееееееее
Hn-1L! ?a
(7.6.54)
Using the well-known relations of the G function and factorials discussed
in the previous subsection, we can generalize the result to an arbitrary
7. Fractals
815
order of fractional differentiation by replacing n ! by GHn + 1L. The general
formula follows thus by
+-q
a, x f HxL =
x
1
ееее
ееееее Hx - x0 Lq-1 f Hx0 L ? x0
GHqL ?a
(7.6.55)
with ReHqL > 0.
This kind of operator is denoted as the Riemann (R) version of the
fractional integral by Miller and Ross [7.11]. The Liouville (L) version of
this operator follows if we replace the lower boundary a of the integral by
-╤; that is, +-q
-╤, x f HxL is called the Liouville fractional integral. A
sufficient condition that this integral converges is that f H-xL = oHx-q-e L for
e > 0 and x ь ╤. The special case where a = 0
-q
1
x
ееееее ?0 Hx - x0 Lq-1 f Hx0 L ? x0 ,
+0, x f HxL = ееее
GHqL
q > 0,
(7.6.56)
is known as the Riemann?Liouville (RL) fractional integral. A sufficient
condition that the RL integral converges is given by f H1 Й xL = OHx1-e L for
e > 0. Functions satisfying this relation are called functions of the
Riemann?Liouville type. For example, the functions xa with a > -1 and a
constant belong to this class of functions. We recognize that the different
definitions of Riemann?Liouville fractional integrals differ only in the
lower boundary of the integral. The reader might suppose that this small
difference is of minor importance. The following subsection will
demonstrate that this assumption is not correct. The change of the lower
boundary has very far-reaching consequences in the calculation of
fractional derivatives.
So far, we introduced the notation of the fractional integral. A fractional
derivative is connected with a fractional integral by introducing a positive
order of differentiation in the operator +-q
a, x . This shift of order can be
obtained by introducing an ordinary differentiation followed by a
fractional integration. We thus define a fractional differentiation by
dn
-Hn-sL
f HxL
+sa, x f HxL := I ееее
nе M +a, x
dxееее
n ? 1, s > 0, n - s > 0.
with
(7.6.57)
In this Riemann notation, the fractional derivative depends on a lower
boundary a of the integral. This dependence disappears if we consider only
the RL operator with a = 0.
816
7.6 Fractional Calculus
Up to the present point, we discussed the essentials of the theory of RL
integrals. If we intend to use computer algebra in connection with RL
operators, we need to know how RL operators are implemented. Thus, the
next step is to create a function in Mathematica which carries out the
calculation. We call this function RiemannLiouville[]. Since the RL
integral is applied to functions depending on one independent variable, say
x, we need to supply this information to the function. Another quantity
which must be given by the user is the order of differentiation q. In
addition to these two input variables, we need information on the lower
boundary of the integration interval. Thus, our function needs, in addition
to the function on which we apply the RL operator, three input quantities.
The lower boundary is superfluous if we treat a RL integral. The following
definition of the Riemann?Liouville fractional integral incorporates the
theoretical considerations discussed above:
Remove@RiemannLiouvilleD;
RiemannLiouville@1, 8x_, order_, a_: 0<D :=
Hx aL ^ order Й Gamma@1 orderD;
H main function L
RiemannLiouville@f_, 8x_, order_, a_: 0<D :=
Block@8n, int, y<,
If@NumericQ@orderD && Simplify@order > 0D,
n = Floor@orderD; q = order nD;
int = Integrate@Hx yLq1 Hf Й. x ▒ yL,
8y, a, x<, GenerateConditions ▒ FalseD;
D@int Й Gamma@qD, 8x, n<D Й; FreeQ@int, yD
D
At this stage, we know how functions are treated by a RL integral. Before
we apply RiemannLiouville[] to a mathematical problem or use it in
physical models, we introduce some general properties of the fractional
derivative. These properties are important for manual as well as for
automatic calculations. They also serve to extend the properties of the
function RiemannLiouville[].
7. Fractals
817
7.6.2.1 Properties of Riemann-Liouville Operators
The main properties needed in an implementation of RL operators are
linearity and the composition rule. These two properties are basic
properties in addition to the Leibniz rule of differentiation and the chain
rule. Let us discuss these properties in more detail. In the implementation
of the mathematical properties, linearity and the composition of derivatives
are of importance. The other two relations are of minor practical
importance.
1 Linearity
Linearity is one of the basic properties of a RL operator. This property
guarantees that the superposition of a RL operators applied to different
functions is the same as the application of the RL operator on the
superposition of functions. Linearity of a RL operator means
+sa, x Ha f HxL + b gHxLL = a +sa, x f HxL + b +sa, x gHxL,
(7.6.58)
with a and b as real constants. Relation (7.6.58) is implemented by two
functions. The first function removes common constants from the
argument of the input function:
RiemannLiouville@c_ f_, 8x_, order_, a_: 0<D :=
c RiemannLiouville@f, 8x, order, a<D Й; FreeQ@c, xD;
The second part of the linearity represents a superposition of two
functions. This property is implemented as
RiemannLiouville@f_ + g_, 8x_, order_, a_: 0<D :=
RiemannLiouville@f, 8x, order, a<D +
RiemannLiouville@g, 8x, order, a<D
Both definitions combined represent relation (7.6.58). Linearity of the RL
operator means that the operator +sa, x can be distributed through the terms
of a finite sum; that is,
+sa, x ?ni=0 fi HxL = ?ni=0 +sa, x fi HxL.
(7.6.59)
818
7.6 Fractional Calculus
Another important relation is the composition rule of fractional
differentiation.
2 Composition Rule
In the case of RL integrals for m, n > 0 and f HxL continuous, the relation
-m
-Hm+nL
+0, x +-n
0, x f HxL = +0, x
f HxL
(7.6.60)
holds.
The composition rule combining two fractional derivatives of different
order is
p
+sa, x +a,p x f HxL = +s+
a, x f HxL,
(7.6.61)
with p < 0 and f HxL finite at x = a. This property is another rule to extend
the definition of the function RiemannLiouville[]. The following lines
represent the above relation
RiemannLiouville@ RiemannLiouville@f_,
8x_, order1_, a_: 0<D, 8x_, order2_, a_: 0<D :=
RiemannLiouville@f, 8x, order1 + order2, a<D Й;
order1 < 0
In the case of p > 0, the following relation holds:
p
s+ p
-p
p
+sa, x +a,p x f HxL = +s+
a, x f HxL - +a, x H f HxL - +a, x +a, x f HxLL
(7.6.62)
where the last term is
p-k
+-a,px +a,p x f HxL = f HxL - ?m
,
k=1 ck x
(7.6.63)
with 0 < p ╖ m < p + 1. The constants ck in Eq. (7.6.63) are constants of
integration. In the case of the RL integral Ha = 0L, these constants are given
by
1
p-k
ееееееееее +0, x f HxL ?x=0 .
ck = ееееееееееееееее
GH p-+k+1L
(7.6.64)
The difference of p > 0 or p < 0 can be demonstrated by the example
+1a, x +-1
a, x f HxL = f HxL
for p < 0 and
(7.6.65)
7. Fractals
819
1
+-1
a, x +a, x f HxL = f HxL + c
(7.6.66)
with c a constant. This example also demonstrates the general property that
RL integrals do not commute.
3 Chain Rule
The chain rule of a RL operator is
╤
+qa, x f HgHxLL = ?
j=0
ij q yz
d j f HgHxLL
x j-q
еееее ееееееееееееееее
еееее .
j z ееееееееееееееее
GH1+
j-qL
dx j
k j{
(7.6.67)
The complexity of this result will inhibit its general utility in connection
with computer algebra. The chain rule creates an infinite series that offers
little hope of being expressible in closed form.
4 Leibniz's Rule
The rule for differentiation of a product of two functions is a familiar
result in calculus. It states that
n
d H f HxL gHxLL
ееееееееееееееее
еееееееееее = ?
dxn
n
n d n- j f HxL d j gHxL
jij zyz ееееееее
ееееееее
еее
n- j ееее ееееееееееее
dx j
k j { dx
j=0
(7.6.68)
for non-negative integers n. The generalization of Leibniz's rule to
negative numbers is given by
╤
+qa, x H f HxL gHxLL = ?
j=0
ij q yz q- j
j z +a, x f HxL +a,j x gHxL,
k j{
(7.6.69)
iqy
where the binomial jj zz = GHq + 1L Й HGH j + 1L GHq - j + 1LL is expressed by
k j{
Euler's G function. Again we face the problem that Leinbiz's rule results
into an infinite series. This series may collapse to a simple expression if
the functions f and g are simple. However, in general computer algebra
cannot handle this relation.
The discussed Mathematica code shows that it is sufficient for an
implementation to use the definition given by the RL operator in Eq.
(7.6.56)?(7.6.59). The mathematical formulas and the Mathematica code
above show that the RL operator in mathematical and Mathematica
notation is quite similar. To make this similarity to an identity, we
introduce a special Mathematica notation identical with the RL operator
symbol. The notation +яя, я @яD is connected with the function
820
7.6 Fractional Calculus
RiemannLiouville[]. The template is designed in such a way that it is
identical with the mathematical notation given above. However, this
notation differs somewhat from the standard notation used in the literature.
Since in Mathematica it is safer to handle the lower indices of the operator
+-q
a,x on the right side of the + symbol, we changed the notation given by
for the RL operator. The function
Davis [7.15], who used a +-q
x
RiemannLiouville[] and the template +-q
a,x allow us to carry out different
calculations. The following examples show how the function
RiemannLiouville[] is used and what kind of calculations are supported
by this function.
We note that the following calculations are based on the package
FractionalCalculus` developed by SЭdland and myself. This package is
available from the author by request. To support the future development of
the package FractionalCalculus`, we have to charge the user for the
package.
7.6.2.2 Examples
An example frequently discussed in the literature [7.10, 7.11] is the
differentiation of a constant. From standard calculus, we know that an
ordinary integer differentiation of a constant vanishes. Applying the RL
operator of order q = 1 Й 2 to a numeric constant, say c = 1, we get
+1Й2
0,x @1D
1
ееееееееееееееее
еееееееее!е
Х!!!! Х!!!
p x
This result compared with our knowledge of ordinary calculus is
surprising. Contrary to an ordinary differentiation, the result of a fractional
differentiation does not vanish but depends on the original variable, here x.
The same result follows by applying the function RiemannLiouville[] to
the constant. The difference is that we do not need to specify the lower
boundary. The function RiemannLiouville[] assumes by default that the
lower boundary is zero. However, we can change this boundary value by
providing a third input variable in the second argument of
7. Fractals
821
RiemannLiouville[]. Let us demonstrate this by first using
RiemannLiouville[] with two arguments at the second input position
RiemannLiouville@1, 8x, 1 Й 2<D
Conditions to solve the fractional integral:
x>0
1
ееееееееееееееее
еееееееее!е
Х!!!! Х!!!
p x
The result of both calculations is the same. However, we have the freedom
to choose the lower boundary as a third entry in the function
RiemannLiouville[].
The gained results might contradict the general knowledge that the
differentiation of a constant vanishes. Contrary to the ordinary calculus, in
fractional calculus it is not true that the differentiation of a constant
vanishes. This behavior is obvious if we recall the definition of a fractional
derivative by an integration in Eq. (7.6.56). This nonvanishing of a RL
operator applied to a constant is even true if we allow a general order of
differentiation. Before we can apply the RL operator to the constant, we
have to tell the package FractionalCalculus that we restrict the order of
differentiation to positive values, meaning n > 0. This mathematical
assumption is incorporated into the package FractionalCalculus by the
function Assume[]. This function allows one to specify conditions under
which the integrals are calculated. For our example, we set
Assume@Q > 0D
888Q > 0<, 8Im@QD ▒ 0, Re@QD ▒ Q<<<
This assumption tells the RL operator that n is a positive real number. The
calculation of the RL integral in the general form then gives
822
7.6 Fractional Calculus
d1 = +Q0, x @KD
Conditions to solve the fractional integral:
x > 0 && Re@QD < 1
K xQ
cccccccccccccccc
cccccc
* @1 QD
where K is a constant. The expression shows that for positive n < 1, the RL
operator provides a nonvanishing result containing Euler's G function. A
graphical representation of the result for different n's is given in the
following plot:
Plot3D@d1 Й. K > 1, 8x, .01, 3<,
8Q, 4, 1<, PlotPoints > 40, Mesh > False,
AxesLabel > 8"x", "Q", "+Q0, x @1D"<D;
+n0, x@1D
4
2
0
1
0
-1
-2 n
1
x
2
-3
3 -4
7. Fractals
823
The above calculations show some printings in between the input and
output. These printouts inform you about the conditions under which the
calculation was carried out. The output of conditional information is
controlled by an option of RiemannLiouville[]. The options of the RL
function are
Options@RiemannLiouvilleD
8ShowConditions ▒ True, UniqueSymbols ▒ False,
OldhamSpanierConstants ▒ False,
FractionalIntegrationVariable ▒ y,
ShowFinalResult ▒ False,
ShowLiterature ▒ False, ShowResults ▒ False<
To suppress the information on solution conditions, we set the option
ShowConditions to False.
SetOptions@RiemannLiouville, ShowConditions ▒ FalseD
8ShowConditions ▒ False, UniqueSymbols ▒ False,
OldhamSpanierConstants ▒ False,
FractionalIntegrationVariable ▒ y,
ShowFinalResult ▒ False,
ShowLiterature ▒ False, ShowResults ▒ False<
Now, RiemannLiouville[] does not display any information about the
calculation. An example of a RL integration demonstrates this. The
example uses a power function xm to which we apply the RL operator. Let
us assume that the fractional order of integration is any positive number
greater than zero and let m be a real number. The application of the RL
operator to this function gives
Assume@Q > 0D;
824
7.6 Fractional Calculus
╣
+Q
0, x @x D
x╣+Q * @1 + ╣D
cccccccccccccccc
cccccccccccccccc
* @1 + ╣ + QD
The result is again a power function containing both parameters m and n as
exponents. The behavior of projecting a function into the same class of
function is not typical for the RL operator. The application to other classes
of functions like exponentials, sines, and cosines demonstrates that we get
higher transcendental functions. An example for this behavior is the
function ?a x with a > 0. The application of the RL integral delivers
Assume@D > 0D;
Dx
+Q
D
0, x @ф
фx D DQ J@Q, x DD
cccccccccccccccccccccccccccccccc
cccccccccc
* @QD
which represents the Mittag?Leffler function in Mathematica notation.
The Mittag?Leffler function Ex Hn, aL is defined by
?a x
Ex Hn, aL = ееееaеnееее I1 -
gHn,a xL
ееееееее
еееееееее M.
GHnL
(7.6.70)
Other examples showing the same behavior are the trigonometric functions
+Q
0, x @Sin@Z xDD
81<
1
; ccc
x1+Q Z Fp,q A
c x2 Z2 E
Q
3
Q
4
c
, ccc
c + ccc
c<
81 + ccc
2
2
2
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccccccccccc
ccccc
* @2 + QD
7. Fractals
825
+Q
0, x @Cos@Z xDD
81<
1
; ccc
xQ Fp,q A 1
c x2 Z2 E
Q
Q
4
c + ccc
c , 1 + ccc
c<
8 ccc
2
2
2
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccccccc
* @1 + QD
Both results are connected with hypergeometric functions F p,q . Let us
consider some slightly more complicated functions
f@x_D := HD + xLO
and assume that
Assume@O > 0D;
Then, the fractional integral of this function follows by
+Q
0, x @f@xDD ЙЙ FunctionExpand
x
xQ DO F2,1 @1, O, 1 + Q, ccc
cD
D
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccccccccccc
ccccccc
* @1 + QD
If we change the sign of x in f , we get
I@x_D := HD xLO
+Q
0, x @I@xDD ЙЙ FunctionExpand
x
xQ DO F2,1 @1, O, 1 + Q, ccc
cD
D
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccccccccccc
cccc
* @1 + QD
The difference between the two results is the minus sign in the argument of
F2,1 .
826
7.6 Fractional Calculus
As a another example, let us examine functions containing logarithms. The
fractional integral of lnHxL is given by
+Q
0, x @Log@xDD
xQ HHarmonicNumber@QD + Log@xDL
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccc
* @1 + QD
A more general example is the combination of powers and logarithms by
╣
+Q
0, x @x Log@xDD
1
cccccccccccccccc
cccccccccccccc Hx╣+Q * @1 + ╣D
* @1 + ╣ + QD
HHarmonicNumber@╣D HarmonicNumber@╣ + QD + Log@xDLL
If we combine a power and an exponential, we find a sum of
hypergeometric functions:
╣
+Q
0, x @x Exp@1 Й xDD ЙЙ FunctionExpand
1
S x1+Q Csc@S ╣D F1,1 @1 Q, 2 + ╣, ccc
cD
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccxccccccc * @2 + ╣D * @QD
1
S x╣+Q Csc@S ╣D F1,1 @╣ Q, ╣, ccc
cD
x
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccc
ccc
* @╣D * @1 + ╣ + QD
As a result, a combination of power laws and hypergeometric functions
follows from
2
+Q
0, x @Cos@D xD D
81<
y
i
; x2 D2 Ez
xQ j
z
j1 + Fp,q A 1
Q
Q
c
+
ccc
c
,
1
+
ccc
c
<
8 ccc
{
k
2
2
2
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccccccccccc
ccccc
2 * @1 + QD
7. Fractals
827
A completely different result follows for rational functions. First, let us set
the integration order to the special value -1 Й 2. For the function
H1 - xL Й H1 + a xL, we find
1x
+1Й2
0, x A ccccccccccccccccc E
1+Dx
Х!!!! Х!!!! Х!!!!!!!!!!!!!!!!
Х!!!! Х!!!!
2 I x D 1 + x D + H1 + DL ArcSinhA x D EM
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccc
Х!!!!!!!!!!!!!!!!!!!!
D3Й2 S + S x D
The result contains hyperbolic functions. For arbitrary n, we find
1x
hyper = +Q
0, x A ccccccccccccccccc E ЙЙ FunctionExpand
1+Dx
Conditions to solve the fractional integral:
1
1
1
x > 0 && Re@QD > 0 && J ccccccccc √ 0 ╩╩ 1 + ccccccccc ├ 0 ╩╩ ImA ccccccccc E ° 0N
xD
xD
xD
x1+Q F2,1 @1, 2, 2 + Q, x DD
xQ F2,1 @1, 1, 1 + Q, x DD
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccccccccc cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccccccccccccc
* @1 + QD
* @2 + QD
If we choose n = 1 Й 2, the result reduces to the previous result:
1
hyper Й. Q ▒ cccc ЙЙ Simplify
2
Х!!!! Х!!!! Х!!!!!!!!!!!!!!!!
Х!!!! Х!!!!
2 I x D 1 + x D + H1 + DL ArcSinhA x D EM
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccc
Х!!!!!!!!!!!!!!!!!!!
!
D3Й2 S + S x D
The following plot shows the result of the fractional derivative where x
and n are used as coordinates and a as a changing parameter. The static
picture shows the transition to the value at n = 1 Й 2. In addition, the
variation of a visualizes the change of the surface. We observe that an
increase in a will stretch out the surface to a more or less flat plane:
828
7.6 Fractional Calculus
a = 0.1
1
0.5
0
1-x
-0.5
+-n
0, x @ еееееееееееееееееееее D -1
1+ax
-1.5 2
1.5
3
1
n 0.5
2
1 x
Including hyperbolic functions as arguments for the RL operator, we find
╣
Dx
+Q
D ЙЙ FunctionExpand
0, x @x Sinh@J xD ф
1
cccccccccccccccc
cccccccccccccccccc
2 * @1 + ╣ + QD
Hx╣+Q * @1 + ╣D HF1,1 @1 + ╣, 1 + ╣ + Q, x HD JLD F1,1 @1 + ╣, 1 + ╣ + Q, x HD + JLDLL
The pure Sinh with a square root of the independent variable as argument
in the RL integral reduces to
1Й2
+Q
DD ЙЙ FunctionExpand
0, x @Sinh@x
2 ccc2c +Q
1
Х!!!! ccc12c + ccc12c
S x
1
I ccc
2c QM+Q
I ccc1c
2
Х!!!!
xE
H1+2 QL A
The result is a Bessel function of I type multiplied by a power function.
Even if we look at special functions like the Bessel functions, we can
calculate the RL integral. The following example takes a Bessel J as
argument in the RL integral:
7. Fractals
829
+Q
0, x @BesselJ@n, xDD ЙЙ FunctionExpand
1
n
n
c + ccc
c , 1 + ccc
c<
8 ccc
2
2
2
2n xn+Q Fp,q A
Q
ccc
c,
2
n
ccc
c
2
2
Q
ccc
c<
2
x
; ccccc
E
4
81 + n,
+
+
1+
+
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccccccccccccc
* @1 + n + QD
1
ccc
c
2
n
ccc
c
2
The result of this calculation is a hypergeometric function of general F p,q
type multiplied by a power function. Combining a Bessel functions with a
power, we get
╣
+Q
0, x @x BesselJ@n, xDD ЙЙ FunctionExpand
i
j
n n+╣+Q
j
* @1 + n + ╣D
j
j2 x
k
Fp,q A
╣
╣
1
n
n
c + ccc
c + ccc
c , 1 + ccc
c + ccc
c<
8 ccc
2
2
2
2
2
81 + n,
1
ccc
c
2
+
n
ccc
c
2
+
╣
ccc
c
2
+
Q
ccc
c,
2
1+
n
ccc
c
2
+
╣
ccc
c
2
+
Q
ccc
c<
2
x2 y
z
; ccccccc Ez
zЛ
4 z
{
H* @1 + nD * @1 + n + ╣ + QDL
Again, we find a hypergeometric function Fq, p multiplied by an extended
Х!!!!
power function. A semifractional derivative of 1 К x is given by
1
+1Й2
E
0, x A cccccccccc
Х!!!!
x
0
Surprisingly, this differentiation vanishes. The reason why this result
occurs is obvious from the more general derivative
830
7.6 Fractional Calculus
1
+Q0, x A cccccccccc E
Х!!!!
x
Х!!!! ccc12c Q
S x
cccccccccccccccc
cccccccc
1
* @ ccc
c QD
2
We see that if n = 1 Й 2, the G function approaches infinity and, thus, the
overall behavior is reduced to zero.
The above examples serve to demonstrate that the function
RiemannLiouville[] is designed in such a way that a large class of
function is accessible via integration and differentiation. We already
observed that the application of the RL operators deliver extraordinary
results for simple functions. How these results are useful in connection
with physical applications is discussed in Section 7.6.4.
7.6.3 Mellin Transforms
Frequently in mathematical physics we encounter pairs of functions related
by an expression of the form
b
gHxL = ?a f HkL KHx, kL ? k.
(7.6.71)
The function gHxL is called the integral transform of f HkL by the kernel
KHx, kL.
One of the most useful of the infinite number of possible transforms is the
Fourier transform given by
1
2p
╤
еееее ?-╤ f HkL ei x k ? k.
gHxL = ееееееее
Х!!!!!!!!
(7.6.72)
Two modifications of this transformation are the Fourier cosine and the
Fourier sine transforms
?
ееее % ?
g HxL = $%%%%%%
╤
2
ееееp %
gc HxL = $%%%%%%
s
2
p
f HkL cosHx kL ? k
0
(7.6.73)
╤
f HkL sinHx kL ? k
0
(7.6.74)
7. Fractals
831
The Fourier transform is based on the kernel ei x k and its real and
imaginary parts taken separately, cosHk xL and sinHk xL, because these
kernels are the functions used to describe waves. Fourier transforms
appear frequently in studies of waves. The output of a stellar
interferometer, for instance, involves a Fourier transform of the brightness
across a stellar disk. The electron distribution in an atom can be obtained
from a Fourier transform of the amplitude of scattered X-rays.
Three other useful kernels in defining integral transforms are e-k x ,
k Jn Hk xL, and k x-1 . These give rise to the following transformations:
╤
gHxL = ?0 f HkL e-k x ? k
(7.6.75)
defining the Laplace transform,
╤
gHxL = ?0 f HkL k Jn Hk xL ? k,
(7.6.76)
known as the Hankel transform, and
╤
gHxL = ?0 f HkL k x-1 ? k,
(7.6.77)
the Mellin transform. Clearly, the possible types are unlimited. The
following subsection will outline the Mellin transform in more detail.
7.6.3.1 Definition of the Mellin Transform
This subsection is concerned with the theory and application of the Mellin
transform. We define the Mellin transform and its inverse. Several
examples and the basic operational properties of the Mellin transform are
discussed. Historically, Riemann in 1876 [7.16] first recognized the Mellin
transform in his famous memoir on prime numbers. Its explicit formulation
was given by Cahen in 1894 [7.17]. Almost simultaneously, Mellin, in two
papers from 1896 and 1902 [7.18, 7.19], gave an elaborate discussion of
the Mellin transform and its inversion formula.
In this subsection, we study the Mellin transform, which, although closely
related to the Fourier transform, has its own peculiar uses. In particular, it
turns out to be a most convenient tool for solving fractional integral
equations. We recall first that the Fourier transform pair can be written in
the form
832
7.6 Fractional Calculus
╤
FHwL = ?-╤ eiwt f HtL ? t,
with a < ImHwL < b
(7.6.78)
and
1
f HtL = ееее
ееее
2p
╤
?-╤ e-iw t FHwL ? w,
a < g < b.
(7.6.79)
The Mellin transform and its inverse follow if we introduce the variable
changes p = i w, x = et , and fHtL = f HlnHtLL, so that Eq. (7.6.78) and
(7.6.79) become
╤
4H f HtLL = FH pL = ?0 t p-1 f HtL ? t
(7.6.80)
and
1
c+i ╤
еееее
t- p FH pL ? p,
4-1 HFH pLL = f HtL = ееее
2 p i ?c-i ╤
(7.6.81)
respectively. Equation (7.6.80) is the Mellin transform and, (7.6.81) is the
inversion formula for the Mellin transform. The transform normally exists
only in the range a < ReH pL < b, and the inversion contour must lie in this
strip.
The following theorem collects the main properties of the Mellin transform.
Theorem: Properties of Mellin Transform
If 4H f HtLL = FH pL, then the following properties hold:
7. Fractals
833
No. Property
iL
Scaling
1
4H f Ha tLL = ееее
ее FH pL, a > 0
ap
iiL
Shifting
4Hta f HtLL = FH p + aL
iiiL
Derivatives
4H f HnL HtLL =
GH pL
H-1Ln ееееееее
ееееее е FH p - nL
GH p-nL
ivL
Derivative multiplied 4Htn f HnL HtLL = H-1Ln ееееееее
еееееееее FH pL
GH pL
with a power
vL
Differential operator
GH p+nL
n
d
е L f HtLM =
4IHt ееее
dt
H-1Ln pn FH pL
t
FH p+1L
vIL
Integrals
4I?0 f HuL ? uM = - ееееееееpеееееееее
viiL
nth repeated Integral
4HIn f HtLL =
t
4I?0 In-1 f HuL ? uM =
GH pL
H-1Ln ееееееее
еееееееее FH p + nL
GH p+nL
viiiL Convolution type I
4H f HtL * gHtLL =
╤
4H?0 f HuL gH ееutее L ееее1u ? uL =
FH pL GH pL
ixL
Convolution type II
4H f HtL КgHtLL =
╤
4H?0 f Ht uL gHuL ? uL =
FH pL GH1 - pL
In this table, In f HtL denotes the nth repeated integral if f HtL defined by
t
In f HtL = ?0 In-1 f HuL ? u. Ф
The package FractionalCalculus contains a function MellinTransform[], which is accessible by the template 4яя @яD, where the lower
placeholder represents the original variable and the upper placeholder
represents the Mellin variable. The placeholder in [] contains the function
which is transformed. The following examples demonstrate the application
of the Mellin transform to different functions.
834
7.6 Fractional Calculus
7.6.3.2 Examples for Mellin Transforms
Before we discuss specific examples and applications of the Mellin
transform, let us demonstrate some general properties. The scaling
property of the Mellin transform for an arbitrary function f is given by
Remove@f, ID;
Assume@O > 0D;
4pt @f@O tDD
Op 4pt @f@tDD
The result is identical with property i) of the above table. The shifting
property follows from
4pt @tO f@tDD
4p+O
t @f@tDD
The following relations demonstrate that the Mellin transform is defined
for powers:
4pt @f@tO DD
p
ccc
c
4tO @f@tDD
cccccccccccccccc
cccccccccc
O
for rational functions:
7. Fractals
835
1
f@ ccc
cD
t
4pt A ccccccccccccccc E
t
41p
t @f@tDD
and for logarithms:
4pt @Log@tD f@tDD
MellinTransformH0,0,1L @f@tD, t, pD
Even general derivatives
MellinTransform[]:
can
be
handled
by
the
function
4pt @≥t f@tDD ЙЙ FunctionExpand
H1 pL 41+p
@f@tDD
t
4pt @≥t,t f@tDD ЙЙ FunctionExpand
H2 + pL H1 + pL 42+p
@f@tDD
t
The results are special cases of the general formula from above. The
Mellin transform of an integral is given by
t
4pt A? f@WD е WE ЙЙ FunctionExpand
0
41+p
t @f@tDD
ccccccccccccc
cccccccccccccccc
p
The convolution properties viii) and ix) are
836
7.6 Fractional Calculus
t
┬
g@ ccc
cD
W
4pt A? f@WD ccccccccccccccc е WE
W
0
4pt @f@tDD 4pt @g@tDD
or
┬
4pt A? f@t WD g@WD е WE
0
4pt @f@tDD 41p
t @g@tDD
These general properties are important in the treatment of the following
applications. Before we discuss the capabilities of the Mellin transform in
connection with integrals, integral equations, and differential equations, we
demonstrate the application of the Mellin transform to special functions.
The first example is concerned with the function f HtL = e-n t with n > 0.
The Mellin transform of the exponential function follows by applying the
operator 4яя @яD to the function
4pt @Exp@n tDD
np * @pD
This result is characteristic for an exponential function. In the Mellin
space, this kind of function is represented by the G function divided by n to
the power of p denoting the factor in the exponent. The function
MellinTransform[] also tells us that the real part of n and the real part of
p must be greater than zero. Another example of interest is given by the
rational function 1 Й H1 + tL. The Mellin transform of this function is
7. Fractals
837
1
4pt A cccccccccccc E
1+t
S Csc@p SD
The Mellin transform of the generalized expression f HtL = 1 Й H1 + tLn
follows from
1
4pt A cccccccccccccccccccc
E
H1 + tLn
* @n pD * @pD
cccccccccccccccc
cccccccccccccccccc
* @nD
The result is represented by a fraction of G functions depending on the
Mellin variable p and on the exponent n. The representation of the Mellin
transform in terms of G functions is very useful in connection with the
solution of fractional differential equations. Another interesting example
containing an exponential function is the Mellin transform of the function
f HtL = 1 Й Het ■ 1L. The two Mellin transforms read
1
4pt A cccccccccccccccc
cccccccccc E
Exp@tD 1
* @pD Zeta@pD
The result contains a special function the so-called Riemann z function.
The second representation of f HtL with the - sign replaced by the + sign
gives
1
4pt A cccccccccccccccc
cccccccccc E
Exp@tD + 1
2p H2 + 2p L * @pD Zeta@pD
838
7.6 Fractional Calculus
Here again, the G function and the z function are involved in the
representation of the Mellin transform. An example containing
trigonometric functions is
4pt @Sin@H1 tLDD
pS
2
* @pD SinA1 ccccccccc E
The result contains trigonometric as well as the G function. The Mellin
transform of the pure Cos[] is given by
4pt @Cos@Z tDD
pЙ2
HZ2 L
pS
CosA ccccccccc E * @pD
2
where w is a positive constant. Other special functions are logarithms. An
example containing a logarithmic expression is given by
4pt @Log@1 + tDD
S Csc@H1 + pL SD * @pD
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc
* @1 pD
These few examples demonstrate that the Mellin transform of special
functions can be calculated in a direct way. We note that the package
FractionalCalculus is capable to calculate all the Mellin transforms and
more tabulated by Oberhettinger [7.20].
The inverse of the Mellin transform (InverseMellinTransform[]) in the
я
package FractionalCalculus is accessible by the operator H4-1 Lя @яD. The
subscript denotes the Mellin variable and the superscript denotes the
original variable. The template of the inverse Mellin transform is
connected with the function InverseMellinTransform[] . A simple
example for an inversion is
7. Fractals
839
t
H41 Lp @Gamma@pDD
фt
which just delivers the exponential function. Another simple example is
t
H41 Lp @Gamma@p + nDD
фt tn
where n is a positive number. More complicated results follow from
p
p
t
H41 Lp AGamma@1 + pD GammaA ccccc E GammaA1 ccccc EE
S
S
/
2,1
1,2 At
/
?
1
?
c << ╩
8<
880, ccc
?
S
?
?
E
?
?
1
?
880,
ccc
c
<,
81,
1<<
╩
8<
?
?
S
2,1
where
1,2 represents a generalized hypergeometric function, so called
Fox functions. A similar result follows from
t Gamma@1 + pD
H41 Lp A cccccccccccccccc
cccccccccccccccc E
Sin@pD
?
1
?
c << ╩
8<
880, ccc
?
S
?
?
E
?
?
1
?
880,
ccc
c
<,
81,
1<<
╩
8<
?
?
S
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccc
S
/
2,1
1,2 At
If we combine a G function and a trigonometric function in the Mellin
space by a product, we find
840
7.6 Fractional Calculus
t
H41 Lp @Gamma@pD Sin@pDD ЙЙ FunctionExpand
S
?
1
?
c <<
8< ╩
880, ccc
?
S
?
?
E
At
?
1,2
?
1
?
c <<
?
? 880, 1<< ╩ 880, ccc
S
/
1,0
Another rational expression of G functions and the Sin[] gives
Gamma@1 + pD Sin@pD
t
H41 Lp A cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccc E
Gamma@pD Gamma@2 pD
S
/
1,0
3,2 At
?
1
?
c <<
8< ╩
880, 1<, 80, 2<, 80, ccc
?
S
?
?
E
?
?
1
?
880, ccc
c <<
?
? 881, 1<< ╩
S
These small selection of special combinations of G functions demonstrate
that the inverse Mellin transform is a powerful tool to represent special
functions. The package FractionalCalculus allows one to calculate a large
number of special functions, including Fox's H function, which is a
generalization of the Meijer G function. The following applications
demonstrate how a Mellin transform can be used to solve specific
mathematical and physical problems.
7.6.3.3 Solution of Integrals
Let us first discuss the solution of specific integrals of the form
t
F@tD == ? f@WD е W;
0
We first apply the Mellin transform on both sides of the equation:
7. Fractals
841
t
r1 = 4tp AF@tD == ? f@WD е WE
0
* @pD 41+p
t @f@tDD
4pt @F@tDD == cccccccccccccccccccccccccccccccc
cccccccccccccc
* @1 pD
The result represents the solution of the integral in Mellin space. The
inversion of the Mellin transform provides us with the integral value:
t
H41 Lp @r1D
1+p
t * @pD 4t @f@tDD
F@tD == H41 Lp A cccccccccccccccccccccccccccccccc
cccccccccccccc E
* @1 pD
under the condition that ≥t F = f HtL and FH0L = 0. An integral satisfying
t
these conditions is given by ?0 cosHtL dt. The Mellin transform according
to the above formula gives for the integrand
intM = 4p+1
t @Cos@tDD *@pD Й *@1 pD ЙЙ FullSimplify
pS
2
* @pD SinA ccccccccc E
Since the inversion of the Mellin transform is essentially based on G
functions, we first have to represent the trigonometric function by G
functions. The package FractionalCalculus contains general
transformation rules to carry out this transformation. Applying the rules
TrigToGammaRules to the result intM, we find
intM = intM Й. TrigToGammaRules
S * @pD
cccccccccccccccc
cccccccccccccccc
ccccccc
p
p
* @1 ccc
c D * @ ccc
cD
2
2
containing only G functions. The inverse Mellin transform now follows by
842
7.6 Fractional Calculus
t
H41 Lp @intMD ЙЙ FunctionExpand ЙЙ PowerExpand
1
FoxH::changedstructure : Warning: cccccccc
cccccc FoxH@
Х!!!!
2 S
1
1
1
t
88<, 8<< , 999 cccc , cccc ==, 990, cccc === , cccc , 81, 0, 0, 2< D:
2
2
2
2
This Fox Hfunction has a changed structure in comparison
with the input. Please check your input data.
Sin@tD
The inverse Mellin transform is based on the definition of Fox's H
function. This connection is always used by FractionalCalculus to reduce
the result to a special function. The direct integration using Mathematica
provides the same result:
t
? Cos@WD е W
0
Sin@tD
Another integral also satisfying the above conditions is given by
t
WD
ccccccccc е W ЙЙ Timing
? cccccccccccccccc
E
0 1 Hb WL
96.81 Second, IfAt > 0 && Re@DD > 1 && Re@ED > 0,
1+D
1+D+E
t
t1+D F2,1 @1, ccccccc
c , cccccccc
cccc , bE tE D
WD
E
E
cccccccc
c е WE=
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccc , ? cccccccccccccccc
E
1+D
0 1 Hb WL
Let us assume that the parameters a, b, and b are positive constants:
Assume@D > 0D; Assume@E > 0D; Assume@b > 0D;
The Mellin transform of the integrand extended by the two G functions
then follows as
7. Fractals
843
vh =
tD
4tp+1 A cccccccccccccccc
ccccccccc E *@pD Й *@1 pD Й. TrigToGammaRules ЙЙ
1 Hb tLE
Simplify ЙЙ Timing
1+p+D
1+p+DE
b1pD S * @pD * @ cccccccc
cccc D * @ cccccccc
cccccccc D
E
E
91.1 Second, cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccccccccc =
22 p2 D+E
2+2 p+2 D+E
cccccccc D * @ cccccccccccccccc
cccccc D
E * @1 pD * @ cccccccccccccccc
2E
2E
We represent the result by G functions because the inversion of the Mellin
transform relies on this functions. The inversion of the Mellin transform
gives
t
H41 Lp @vhP2TD ЙЙ Timing
i
1,2
1j
jb1D S
91.37 Second, cccc j
3,3 A
Ej
k
?
1+D
1
2+2 D+E
1
? 881, 1<, 8 ccccccc
c , ccc
c << ╩
88 cccccccc
ccccccc , ccc
c <<
y
?
z
E
E
2E
E
?
bt?
Ez
z
?
z=
?
1+D
1
2+2 D+E
1
?
?
88 ccccccc
c
,
ccc
c
<<
╩
880,
1<,
8
cccccccc
c
ccccc
c
,
ccc
c
<<
?
E
E
2E
E
{
/
representing the result in terms of a Fox function. The direct integration
with integrate has a different representation by hypergeometric functions
t
WD
ccccccccc е W
? cccccccccccccccc
E
0 1 Hb WL
IfAt > 0 && Re@DD > 1 && Re@ED > 0,
1+D
1+D+E
t
t1+D F2,1 @1, ccccccc
c , cccccccc
cccc , bE tE D
WD
E
E
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccc , ? cccccccccccccccc
cccccccc
c е WE
E
1+D
0 1 Hb WL
Another application of the Mellin transform is the calculation of the
moments of the Kohlrausch?William?Watts (KWW) distribution given by
KWW@x_D := фHb xL
E
844
7.6 Fractional Calculus
The moments of this distribution are given by
t
D
? W KWW@WD е W
0
S
IfAt > 0 && Re@ED > 0 && E H ? Arg@bD ?L < cccc ,
2
1+D
cccccccc
1+D
t
c , bE tE D
HbE L E J@ ccccccc
E
E
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc , ? фHb WL WD е WE
E
0
where a, b, and b are positive constants.
Assume@D > 0D; Assume@E > 0D; Assume@b > 0D;
The Mellin transform of the integrand follows by
intM = 4Wp+1 @WD KWW@WDD *@pD Й *@1 pD
1+p+D
b1pD * @pD * @ cccccccc
cccc D
E
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc
E * @1 pD
The inversion shows the coincidence with the direct method:
t
res = H41 Lp @intMD ЙЙ PowerExpand ЙЙ Simplify
1+D
b1D J@ ccccccc
c , bE tE D
E
cccccccccccccccccccccccccccccccc
cccccccccccccccc
E
7.6.3.4 Integral Equations
Another application of the Mellin transform is the solution of integral
equations of the convolution type. Let us consider the general form of a
first-kind Fredholm convolution integral equation. The unknown function
in this equation is denoted by f :
7. Fractals
845
┬
firstFredholm = ? f@WD k@t WD е W == g@tD
0
┬
? f@WD k@t WD е W == g@tD
0
k and g are the kernel and the inhomogenity of the equation. If g equals
zero, we have a homogenous integral equation of Fredholm type. The
Mellin transform applied to this equation gives
MellinFirstFredholm = 4pt @firstFredholmD
p
p
41p
t @f@tDD 4t @k@tDD == 4t @g@tDD
If we replace p by 1 - p in the above expression, we find a standardized
representation of the Fredholm equation in Mellin space:
standard = MellinFirstFredholm Й. p ▒ p + 1
1p
4pt @f@tDD 41p
t @k@tDD == 4t @g@tDD
which is solved by
solMellin = Solve@standard, 4tp @f@tDDD
41p
t @g@tDD
994pt @f@tDD ▒ cccccccccccccccc
ccccccccccccc ==
41p
t @k@tDD
The inversion of the Mellin transform gives us the formal solution of the
integral equation:
846
7.6 Fractional Calculus
t
H41 Lp @solMellinD
1p
t 4t @g@tDD
ccccccccccccc E==
99f@tD ▒ H41 Lp A cccccccccccccccc
41p
t @k@tDD
The second type of convolution-type integral connected with a Mellin
transform is given by the equation
t
┬
cD
k@ ccc
W
secondFredholm = g@tD == ? f@WD ccccccccccccccc е W
W
0
g@tD == ?
┬
0
t
f@WD k@ ccc
cD
W
cccccccccccccccc
cccccccc
cccc е W
W
Again, k and g are the kernel and the inhomogenity, respectively. For this
second kind of convolution equation, a Mellin transform provides
MellinSecondFredholm = 4pt @secondFredholmD
4pt @g@tDD == 4pt @f@tDD 4pt @k@tDD
We realize that for the second kind of equation, we do not need to shift the
Mellin variable p in any way. Thus, we can proceed to solve the resulting
relation to derive the solution in Mellin space:
solMellin = Solve@MellinSecondFredholm, 4pt @f@tDDD
4pt @g@tDD
994pt @f@tDD ▒ cccccccccccccccc
ccccccccc ==
4pt @k@tDD
The inversion of the relation gives the formal solution in original variables:
7. Fractals
847
t
H41 Lp @solMellinD
p
t 4t @g@tDD
ccccccccc E==
99f@tD ▒ H41 Lp A cccccccccccccccc
4pt @k@tDD
Thus, the algorithmic procedure to solve the two integral equations of
convolution type must distinguish two cases of specific kernels. The
characteristic is even more pronounced in the Mellin space, where the two
cases differ by shifts in the Mellin variable from each other. A function
which solves first Fredholm equations of convolution type has to be
sensitive on this case. The following function realizes an automatic
solution procedure for the two types of integral equation:
Clear@ISolveFirstFredholmD
ISolveFirstFredholm@equation_, depend_, independ_D :=
Block@8mtr, solmtr, p, k, vh, solexp<,
mtr = MellinTransform@equation, independ, pD;
vh = k == First@Cases@
Level@mtr, Depth@mtrDD, a_. MellinTransform@
Apply@depend, 8independ<D, t_, p_D :> pDD;
solexp = Solve@vh, kD Й. k ▒ p ЙЙ Flatten;
mtr = mtr Й. solexp;
solmtr = Solve@mtr, MellinTransform@
Apply@depend, 8independ<D, independ, pDD;
InverseMellinTransform@solmtr, p, independDD
The above lines carry out first the Mellin transform of the integral
equation. In a second step, the Mellin variable for the unknown function is
determined. If a shift in the Mellin variable occurs, this shift is eliminated
by an appropriate transformation. Next, the solution in Mellin space is
calculated. The last step transforms the solution in Mellin space to the
original variables. The general integral equations are solved automatically
by
848
7.6 Fractional Calculus
ISolveFirstFredholm@firstFredholm, f, tD
1p
t 4t @g@tDD
99f@tD ▒ H41 Lp A cccccccccccccccc
ccccccccccccc E==
41p
t @k@tDD
and the solution of the second integral equation follows from
ISolveFirstFredholm@secondFredholm, f, tD
p
t 4t @g@tDD
99f@tD ▒ H41 Lp A cccccccccccccccc
ccccccccc E==
4pt @k@tDD
Thus, we have a general procedure to solve first-kind Fredholm integral
equations of the convolution type. A special example of the first
convolution type is given by
┬
1
equation1 = ? Sin@t WD f@WD е W == cccccccccccccccccccc
H1
+
tLn
0
┬
n
? f@WD Sin@t WD е W == H1 + tL
0
where the kernel is given by a trigonometric function. The solution of this
integral equation then follows by
7. Fractals
849
solution1 =
ISolveFirstFredholm@equation1, f, tD ЙЙ FunctionExpand
1
1
1
FoxH::changedstructure : Warning: cccc
cccccc FoxH@ 999 cccc , cccc ==, 8<= ,
Х!!!!
2
2
S
1
1
999 cccc , cccc =, 81 + n, 1<=, 8<= , t , 82, 1, 1, 2< D:
2
2
This Fox Hfunction has a changed structure in comparison
with the input. Please check your input data.
99f@tD ▒
1
n
1
n
1
1n
1
cccccccccccccc JHг tL ccc2c + ccc2c Hг tL ccc2c + ccc2c Cos@tD CscAJ cccc + cccccccccccc N SE
* @nD
2
2
1n
n
1
iХ!!!!!!!!!!!
CscAJ cccccccccccc + cccc N SEN + cccccccccccccccccccccccccccccccc
cccccccccccccc j
j г t
2
2
H2 + nL H1 + nL S k
1
1
1
n
Х!!!!!!!!
г t CscAJ cccc + cccc H1 + nLN SE CscAJ cccc + cccc N SE
2
2
2
2
81<
t2
y
Fp,q A 3
; ccccccc E Sin@n SDz
z+
n
n
4
c ccc
c , 2 ccc
c<
8 ccc
{
2
2
2
n
n
1
n
1
1+ ccc
c
1+ ccc
c
2
2
Hг tL
CscAJ cccc cccc N SE
cccccccc
cccccccccccc JHг tL
2
2
t3 * @nD
1
n
CscAJ cccc H1 + nL cccc N SE Sin@tDN==
2
2
An example for the second convolution type integral equation is given by
the equation:
┬
Cos@t WD
equation2 = 0 == Exp@tD ? I@WD cccccccccccccccc
ccccccc е W
W
0
0 == фt ?
┬
0
Cos@t WD I@WD
cccccccccccccccc
cccccccccccccccccccc е W
W
Again, we replaced the kernel by a trigonometric function. The solution of
this equation follows from
850
7.6 Fractional Calculus
solution2 = ISolveFirstFredholm@equation2, I, tD
FoxH::changedstructure :
1
1
1
Warning: cccccccc
cccccc FoxH@ 999 cccc , cccc ==, 8<= ,
Х!!!!
2
2
2 S
1
1
999 cccc , cccc ==, 8<= , t , 81, 1, 1, 1< D:
2
2
This Fox Hfunction has a changed structure in comparison
with the input. Please check your input data.
2t
99I@tD ▒ cccccccccccccccc
ccccccccc ==
S H1 + t2 L
Another example is concerned with the Laplace integral equation
┬
1
equation3 = ? фt W f@WD е W == cccccccccccccccccccc
H1 + tLn
0
┬
t W
f@WD е W == H1 + tLn
? ф
0
which has the solution
solution3 = ISolveFirstFredholm@equation3, f, tD
фt t1+n
99f@tD ▒ cccccccccccccccc
ccccc ==
* @nD
Replacing in equation3 the exponential constant E by an arbitrary number
a, we get the equation
┬
1
equation4 = ? at W f@WD е W == cccccccccccccccccccc
H1 + tLn
0
┬
t W
f@WD е W == H1 + tLn
? a
0
The solution of this integral equation is
7. Fractals
851
solution4 = ISolveFirstFredholm@equation4, f, tD
at Log@aD Ht Log@aDL1+n
99f@tD ▒ cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccccccccccc ==
* @nD
The second kind of Fredholm equations allows the occurrence of the
unknown function outside of the integral. One of the two standard forms of
the convolution-type Fredholm integral equations of the second kind is
given by
┬
secondFredholm1 = f@tD == g@tD + ? k@t WD f@WD е W
0
┬
f@tD == g@tD + ? f@WD k@t WD е W
0
This equation also can be solved by means of a Mellin transform.
secondF1Mellin1 = 4tp @secondFredholm1D
p
4pt @f@tDD == 4pt @g@tDD + 41p
t @f@tDD 4t @k@tDD
The application of the Mellin operator to the equation shows that the
Mellin transform of the unknown function f occurs with two different
Mellin variables (i.e., with p and 1 - p). This is also true if we carry out
the Mellin transform on the original equation a second time with the
second Mellin variable chosen as 1 - p:
secondF1Mellin2 = 41p
t @secondFredholm1D
1p
p
1p
41p
t @f@tDD == 4t @g@tDD + 4t @f@tDD 4t @k@tDD
Both transforms are equivalent and are the basis for the solution in Mellin
space following from
852
7.6 Fractional Calculus
solutionMellin =
Solve@8secondF1Mellin1, secondF1Mellin2<,
84pt @f@tDD, 41p
t @f@tDD<D
p
4pt @g@tDD + 41p
t @g@tDD 4t @k@tDD
cccccccccccccccccccccccccccccccc
cccccccccccccccccc ,
994pt @f@tDD ▒ cccccccccccccccccccccccccccccccc
1p
p
1 + 4t @k@tDD 4t @k@tDD
p
1p
41p
t @g@tDD + 4t @g@tDD 4t @k@tDD
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccc ==
41p
t @f@tDD ▒ cccccccccccccccccccccccccccccccc
1p
p
1 + 4t @k@tDD 4t @k@tDD
The first formal solution follows from
t
H41 Lp @solutionMellinP1, 1TD
p
1p
p
t 4t @g@tDD + 4t @g@tDD 4t @k@tDD
cccccccccccccccccccccccccccccccc
cccccccccccccccccc E
f@tD ▒ H41 Lp A cccccccccccccccccccccccccccccccc
p
1 + 41p
t @k@tDD 4t @k@tDD
and the second one from the inversion
t
H41 L1p @solutionMellinP1, 2TD
p
1p
41p
t
t @g@tDD + 4t @g@tDD 4t @k@tDD
f@tD ▒ H41 L1p A cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccc E
1p
p
1 + 4t @k@tDD 4t @k@tDD
The second solution is equivalent with the first solution. This is shown by
replacing p by 1 - p in the second Mellin solution. Applying to the result
the standard Mellin transform, we find
t
H41 Lp @solutionMellinP1, 2T Й. p ▒ 1 pD
p
1p
p
t 4t @g@tDD + 4t @g@tDD 4t @k@tDD
f@tD ▒ H41 Lp A cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccc E
1p
p
1 + 4t @k@tDD 4t @k@tDD
7. Fractals
853
which is identical with the first solution. Thus, in an automatic solution
procedure, we only need to treat one of the solutions in Mellin space. The
second type of a second-kind Fredholm equation is given by
┬
t
f@WD
secondFredholm2 = f@tD == g@tD + ? kA cccccc E ccccccccccccc е W
W
W
0
f@tD == g@tD + ?
┬
0
t
f@WD k@ ccc
cD
W
cccccccccccccccc
cccccccc
cccc е W
W
The Mellin transform of this equation shows that the Mellin representation
of the unknown function occurs at both places with the same Mellin
variable p:
4pt @secondFredholm2D
4pt @f@tDD == 4pt @g@tDD + 4pt @f@tDD 4pt @k@tDD
This indicates that the solution procedure of the first-kind Fredholm
equations can be applied to this type of convolution equation. The formal
solution follows from
ISolveFirstFredholm@secondFredholm2, f, tD
t
4pt @g@tDD
99f@tD ▒ H41 Lp A cccccccccccccccc
ccccccccccccccccc E==
1 4pt @k@tDD
Thus, the second type of Fredholm equations can be automatically solved
by the following function:
854
7.6 Fractional Calculus
Clear@ISolveSecondFredholmD
ISolveSecondFredholm@equation_, depend_,
independ_D := Block@8mtr, solmtr, p, k, vh, solexp<,
mtr = MellinTransform@equation, independ, pD;
vh = Complement@
Union@Cases@Level@mtr, Depth@mtrDD,
a_. MellinTransform@Apply@depend, 8independ<D,
t_, p_D :> pDD, 8p<D;
If@Length@vhD >= 1,
mtr1 = Map@
MellinTransform@equation, independ, #D &, vhD;
solmtr = Solve@Flatten@8mtr, mtr1<D,
8MellinTransform@Apply@depend, 8independ<D,
independ, pD, MellinTransform@Apply@depend,
8independ<D, independ, 1 pD<D ЙЙ Flatten;
solmtr = Cases@solmtr,
Rule@MellinTransform@eq_, t_, pD, y___D :>
Rule@MellinTransform@eq, t, pD, yDD,
solmtr = Solve@mtr,
MellinTransform@Apply@depend, 8independ<D,
independ, pDD ЙЙ Flatten;
D;
InverseMellinTransform@solmtr, p, independDD
The formal solution of the second Fredholm equation then follows by
ISolveSecondFredholm@secondFredholm1, f, tD
p
1p
p
t 4t @g@tDD + 4t @g@tDD 4t @k@tDD
9f@tD ▒ H41 Lp A cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccc E=
1p
p
1 + 4t @k@tDD 4t @k@tDD
This solution is actually a formal solution because the inverse Mellin
transform fails to reduce to Fox functions. The main obstacle to prevent
the inversion is the -1 in the denominator preventing a pure representation
by G functions. At this point, we reach the limit of the solution class based
on Fox functions. A specific example demonstrates this behavior more
clearly. Let us examine the Fredholm equation of the second kind:
7. Fractals
855
┬
sF1 = f@tD == H1 + tLD + ? Sin@t WD f@WD е W
0
┬
f@tD == H1 + tLD + ? f@WD Sin@t WD е W
0
The solution should follow by
ISolveSecondFredholm@sF1, f, tD ЙЙ FunctionExpand
9f@tD ▒
Х!!!!!
1
p
p
2p S * @1pD * A ccc
2 * @1 ccc
2c + ccc
2c E * @1+p+DD
2c D * @pD * @p+DD
cccccccccccccccccccccccccccccccc
cccccccccccc + cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccccccccc
* @DD
* @DD
1 t
H4 Lp A cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccc E=
p
p
c
D
+
S
*
@1
ccc
c
D
2 * @1 ccc
2
2
The result demonstrates that the inverse Mellin transform is, in principle,
possible if we extend the meaning of the Barns integral. However, the
resulting function lies outside the class of Fox functions. The second type
of second-kind Fredholm integral equation of convolution type has the
formal solution
ISolveSecondFredholm@secondFredholm2, f, tD
t
4pt @g@tDD
9f@tD ▒ H41 Lp A cccccccccccccccc
ccccccccccccccccc E=
1 4pt @k@tDD
Again, the problem is the same as in the first convolution type. A specific
example shows the problem more clearly:
┬
1
t
f@WD
sF2 = f@tD == cccccccccccccccccccc
+
SinA cccccc E ccccccccccccc е W
?
n
W
W
H1 + tL
0
f@tD == H1 + tLn + ?
┬
0
t
f@WD Sin@ ccc
cD
W
cccccccccccccccc
ccccccccccccccccc
c еW
W
856
7.6 Fractional Calculus
ISolveSecondFredholm@sF2, f, tD
* @npD * @pD
H41 Lp A cccccccccccccccc
ccccccccc1ccccccc
Х!!!!!
pc cEccc E
21+p S * A cccc
c + cccc
t
2 ccccccccc
2
1 cccccccccccccccc
cccccccccccccccc
p
* @1 ccccc D
2
9f@tD ▒ cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccccccc =
* @nD
The occurrence of the -1 in the denominator again prevents a solution by
Fox functions.
At this point, we reach the limits of special functions which serve to solve
the second kind of Fredholm equations. If we are able to enlarge the
definitions of special functions, we will have access to the explicit solution
of the equation. However, so far we did not extend the package
FractionalCalculus to this kind of special functions.
7.6.4 Fractional Differential Equations
The current chapter deals with the formulation and solution of fractional
differential equations (FDEs). We introduce the solution procedure by
recalling the techniques for linear ordinary differential equations (ODEs).
The generalization of these techniques allows us to treat FDEs in different
physical and chemical applications. We discuss relaxation phenomenons in
complex systems like polymers and anomalous diffusion processes.
7.6.4.1 Linear Ordinary Differential Equations
Linear ordinary differential equations (ODEs) occur frequently in
mathematical and physical applications. In general, a differential equation
is an equation that relates an unknown function u and one or more
derivatives or differentials of that unknown function with respect to one or
more independent variables. An ODE contains one or more unknown
functions but depends only on one independent variable. A linear ODE is
an equation containing the dependent variable and its derivatives linearly.
Examples of ODEs are
d uHtL
ееееееее
ееее = f Ht, uHtLL,
dt
(7.6.82)
7. Fractals
857
representing a general first-order ODE for the unknown u. The order of a
differential equation is the order of the highest derivative that appears in
the equation. A linear first-order ODE consists of an equation containing u
linearly. The most general linear first-order ODE is given by
d uHtL
ееееееее
ееее = aHtL uHtL + bHtL,
dt
(7.6.83)
where a and b are real functions of t. This equation is connected with the
Langevin equation if aHtL = -g and bHtL represents a random force.
A general second-order equation is given by the relation
2
d uHtL
d uHtL
ееееееее
ееееее = gIt, uHtL, ееееееее
ееее M.
d t2
dt
(7.6.84)
The most general linear second-order ODE is
2
d uHtL
d uHtL
ееееееее
ееееее = aHtL ееееееее
ееее + bHtL uHtL + kHtL
d t2
dt
with a, b, and k arbitrary functions of t. The next step in increasing the
order is a general nth-order ODE like
n
n-1
d uHtL
d uHtL
d
uHtL
ееееееее
еееееее = hJt, uHtL, ееееееее
ееее , ?, ееееееее
ееееееее
ее N.
d tn
dt
d tn-1
(7.6.85)
So far, we introduced the basic notations to classify ODEs. The question
now is, how can we solve these equations? Before we start to solve the
equations, let us recall the meaning of a solution of ODEs. To say that
u = uHtL is a solution of the differential equation (7.6.85) on an interval K
means that
n
n-1
d uHtL
d uHtL
d
uHtL
ееееееее
еееееее = hJt, uHtL, ееееееее
ееее , ?, ееееееее
ееееееее
ее N
d tn
dt
d tn-1
is satisfied for every choice of t in the interval K. In other words, a
solution, when substituted into the ODE, makes the equation identically
true for t in K. How these solutions, especially for linear ODEs, can be
derived is the subject of the next section. The solution of general nonlinear
ODEs and PDEs is discussed in the book by Baumann [7.21].
858
7.6 Fractional Calculus
7.6.4.2 Solution of ODEs by Integral Transforms
In this subsection, we repeat the solution procedure of linear ODEs by
means of integral transforms. Integral transforms are one of the efficient
methods to solve initial value problems. In detail, we discuss the Laplace
transform technique to solve ODEs. We study this kind of technique
because it is also instrumental in solving fractional differential equations.
One of the key properties of a Laplace transform is the reduction of a
differential equation to an algebraic equation. This property is based on
the transformation of differentials. The result is that an nth-order
derivative f HnL HtL transforms like
/H f HnL HtLL = sn FHsL - ?nm=1 sn-m f Hm-1L H0L.
(7.6.86)
The right-hand side of Eq. (7.6.86) consists of a term containing the
Laplace transform of f , displayed as FHsL, multiplied by the nth power of
the Laplace variable s. The other terms contain the initial conditions
represented by derivatives of f up to order n - 1. We observe that a single
derivative transforms to a polynomial in the Laplace variable s. This
behavior simplifies an ODE to a pure algebraic relation.
The following example demonstrate this for a first-order ODE. The
equation under discussion is the relaxation equation
d f HtL
ееееееее
еееее = - ееее1t f HtL
dt
(7.6.87)
with t, the relaxation time, a constant. Here, we denote the dependent
variable by f to separate the mathematical representation of the equation
from the physical meaning. This equation is represented in Mathematica by
Remove@fD;
1
relaxation = ≥t f@tD == cccc f@tD
W
f@tD
f┘ @tD == ccccccccccccc
W
7. Fractals
859
The Laplace transform of the above equation follows with
lrelax = 3st @relaxationD
3st @f@tDD
f@0D + s 3st @f@tDD == cccccccccccccccc
cccccccc
W
representing an algebraic equation in Laplace space. The Laplace
transform of f is denoted by 3st @ f @tDD. The solution of this equation in
Laplace space follows by solving it with respect to the Laplace transform:
lsol = Solve@lrelax, 3st @f@tDDD ЙЙ Flatten
f@0D
93st @f@tDD ▒ cccccccccccccc
c=
1
s + ccc
c
W
The result shows that the solution in Laplace space is represented by a
function depending on the Laplace variable s and the initial condition
f Ht = 0L. The solution in the original variables results by inverting the
Laplace transform:
t
sol = H31 Ls @3st @f@tDD Й. lsolD
t
?- ееtее f H0L
The solution of the relaxation equation is thus given by an exponential
multiplied by the initial condition f H0L.
This simple example contains the necessary steps to derive a solution for
an initial value problem. We realize that the method presented is
completely algorithmic and can be incorporated into a Mathematica
function. The steps solving a linear initial value problem for an ODE in f
can be summarized as follows:
1. Laplace transform the ODE.
860
7.6 Fractional Calculus
2. Solve the resulting algebraic equation to find the solution in the Laplace
variable.
3. Invert the Laplace transform to find the solution in original coordinates.
These three steps are graphically shown in Figure 7.6.25.
Figure 7.6.25.
Solution procedure based on the Laplace transform for linear ODEs.
We start from a linear ODE D = 0 of arbitrary order. Laplace transform
this equation and solve for the Laplace variable F. The inversion of the
Laplace solution F delivers the solution of the ODE. These steps are
always feasible if the coefficients of the derivatives and the functions are
constants. If we encounter analytic coefficients, we end up with an ODE in
Laplace space.
So far, we demonstrated the solution technique for a homogeneous ODE.
If the equation contains a nonvanishing inhomogeneity, the procedure
works as well. We demonstrate this by extending the relaxation equation
with an inhomogeneity representing an external force, for example. If we
add to the right-hand side of the relaxation equation a term consisting of a
power of t,
7. Fractals
861
1
inHomRelaxation = ≥t f@tD == cccc f@tD +
W
tQ1
i
y
j
j
z
cccccccc z
j cccccccccccccccc
z
Gamma@QD
k
{
f@tD
t1+Q
f┘ @tD == ccccccccccccc + cccccccccccccc
* @QD
W
where n > 0 is a real constant. The Laplace transform of the extended
relaxation equation is
lrelax = 3st @inHomRelaxationD
3st @f@tDD
f@0D + s 3st @f@tDD == sQ cccccccccccccccc
cccccccc
W
Solving with respect to the Laplace variable, we find
lsol = Solve@lrelax, 3st @f@tDDD ЙЙ Flatten
sQ + f@0D
93st @f@tDD ▒ cccccccccccccccc
ccccccccc =
1
s + ccc
c
W
The inversion of this result gives us the solution of the inhomogeneous
relaxation equation:
t
sol = H31 Ls @3st @f@tDD Й. lsolD
t
Q
1
t
ф cccWc H ccc
c L J@Q, ccc
cD
t
W
W
ф cccWc f@0D + cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccc
* @QD
The result is a solution consisting of the homogenous solution and a part
determined by the inhomogeneity. This second part is independent of any
initial condition.
The three steps necessary to solve an initial value problem for ODEs are
incorporated in the function FractalDSolve[]; this function not only
862
7.6 Fractional Calculus
allows the solution of ODEs but is especially designed to solve linear
fractional differential equations. The following line demonstrates the
application of this function to the inhomogeneous relaxation equation:
FractalDSolve@inHomRelaxation, f, tD
9f ▒ FunctionAt,
t
Q
Q
1
1
t
ф cccWc H ccc
c L I* @QD + H ccc
c L f@0D * @QD * @Q, ccc
c DM
W
W
W
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccc
cccccc E=
* @QD
The result is identical to the result derived in the interactive calculation.
The function is also useful in solving linear fractional differential
equations. The following subsection discusses the solution steps necessary.
7.6.4.3 Linear Fractional Differential Equations
Linear fractional differential equations FDEs are integral equations of the
Volterra type. These equations have in common that one part of the
equation consists of an integral operator of Riemann?Liouville or Weyl
type. In general, a FDE is given by
DHt, u, +-n
0, t uL = 0,
(7.6.88)
where n > 0 denotes the order of the FDE. An important property of
(7.6.88) is the linearity of the equation, meaning
DHt, a u + b v, +-n
0, t Ha u + b vLL =
-n
a DHt, u, +-n
0, t u L + b DHt, v, +0, t vL,
(7.6.89)
where a and b are constants and u = uHtL and v = vHtL are functions of the
independent variable t. This property guarantees that the superposition
principle holds and that we can apply integral transforms to solve FDEs.
The solution steps are discussed in the following subsection.
7. Fractals
863
7.6.4.4 Solution of FDEs by Integral Transforms
This section describes a solution procedure of linear FDEs by using
integral transforms. Integral transforms are efficient methods to solve
initial value problems for fractional differential equations. In detail, we
discuss the Laplace and Mellin transform technique to solve FDEs.
One of the key steps in solving FDEs is the Laplace transform as a first
step. This step allows us to reduce a fractional differential equation to an
algebraic equation. We demonstrate this behavior by means of the
generalized relaxation equation:
Remove@fD;
Assume@q > 0D;
1
Frelaxation = +q0,t @f@tDD == cccc f@tD + D
W
f@tD
+qt @f@tDD == D ccccccccccccc
W
where q is a positive number and a is related to the initial condition. The
Laplace transform of the above equation delivers the algebraic equation
lrelax = 3st @FrelaxationD
Conditions to solve the integral:
1 + Re@qD < 0
D
3st @f@tDD
cccccccc
sq 3st @f@tDD == cccc cccccccccccccccc
W
s
The Laplace transform of f is denoted by 3st @ f @tDD. The solution of this
equation in Laplace space follows by solving the above equation with
respect to the Laplace representation of f :
864
7.6 Fractional Calculus
lsol = Solve@lrelax, 3st @f@tDDD ЙЙ Flatten
D
93st @f@tDD ▒ cccccccccccccccc
cccccccc
cc =
1
cL
s Hsq + ccc
W
If we try to apply the inverse Laplace transformation, we end up with an
integral which cannot be solved by Mathematica:
t
H31 Ls @3st @f@tDD Й. lsolD
1
D W1 cccqc ?
t
1+q
Hs W1Йq L
q
Eq,q A Hs W1Йq L E е s
0
However, the resolution of the problem is an additional application of a
Mellin transform to the Laplace representation of the solution. If we, in
addition, shift the Mellin variable, we gain
melEq = 4Vs @lsolD Й. 8V > V + 1, Rule > Equal<
V
SS H1VL
S D W1+ cccqc CscA cccccccc
cqccccccccc E
9* @1 VD 4Vt @f@tDD == cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc =
q
This representation of the solution can be solved for the Mellin transform
of f , providing us with the solution in Mellin space:
smelEq = Solve@melEq, 4Vt @f@tDDD
V
994Vt @f@tDD
SS H1VL
S D W1+ cccqc CscA cccccccc
cqccccccccc E
▒ cccccccccccccccccccccccccccccccc
cccccccccccccccc
ccccccccc ==
q * @1 VD
The inversion of the Mellin transform to the original independent
coordinate t by means of the inverse Mellin transform delivers the solution
in original coordinates:
7. Fractals
865
t
solution = H41 LV @smelEqD ЙЙ PowerExpand ЙЙ Flatten
tq
9f@tD ▒ D W Eq,1 A ccccccc E=
W
The derived solution is given by the Mittag?Leffler function. This example
contains the necessary steps to derive a solution for an initial value
problem. We realize that the method presented is completely algorithmic
and can be incorporated into a Mathematica function. The steps solving a
linear initial value problem for an FDE in f can be summarized as follows:
1. Laplace transform the FDE.
2. Solve the resulting algebraic equation to find the solution in the Laplace
variable.
3. Apply the Mellin transform to find a representation of the Laplace
solution in Mellin space
4. Invert the Mellin transform to find the solution in original coordinates.
These four steps are graphically shown in Figure 7.6.26.
Figure 7.6.26.
Solution steps for FDEs by means of Laplace and Mellin transforms.
866
7.6 Fractional Calculus
The method used is restricted to those functions which can be represented
by the inverse Mellin transform. In other words, the functions must be
given by a Mellin?Barns integral. If this is not the case, the procedure fails
to deliver a solution. However, the solution class derived by this method is
much larger than the solutions derived by a simple Laplace transform.
To summarize the solution procedure, we started from a linear FDE D = 0
of arbitrary order. Laplace transform this equation and solve for the
Laplace variable F. An additional transformation to a Mellin
representation allows us to gain the solution by an inverse Mellin
transform. The inversion of the Mellin solution delivers the solution of the
FDE. These steps are always feasible if the coefficients of the derivatives
and the functions of the FDE are constants.
The three steps necessary to solve an initial value problem for FDEs are
also incorporated in the function FractalDSolve[]; this function not only
allows the solution of ODEs but is especially designed to solve linear
fractional differential equations. The following line demonstrates the
application of this function to the inhomogeneous relaxation equation:
FractalDSolve@Frelaxation, f, tD
q
8f ▒ FunctionAt, D W Eq,1 A Ht W1Йq L EE<
The result is identical to the result derived in the interactive calculation.
7.6.4.5 Fractional Relaxation Equation
Relaxation processes are numerous in physical applications. One of the
famous examples is the decay of a b particle. The temporal behavior of
such a decay is usually described by a first-order ordinary differential
equations. This standard relaxation is also called a Debye process or
Debye relaxation.
7. Fractals
867
1
relaxation = ≥t f@tD + cccc f@tD == 0
W
f@tD
ccccccccccccc + f┘ @tD == 0
W
The solution of this equation follows by applying the function DSolve[] to
the equation
sol1 = DSolve@relaxation, f, tD
t
99f ▒ FunctionA8t<, ф cccWc C@1DE==
The same solution follows by applying the function FractalDSolve[]:
sol2 = -+tf @relaxationD
t
9f ▒ FunctionAt, ф cccWc f@0DE=
Both solutions contain a single constant C@1D and f @0D determining the
initial condition of the relaxation process. The characteristic behavior of a
relaxation process is the exponential decay in time, which is a straight line
in a log plot of the function
868
7.6 Fractional Calculus
LogPlot@f@tD Й. sol2 Й. 8f@0D ▒ 1, W ▒ 1 Й 2<,
8t, .001, 10<, PlotStyle ▒ RGBColor@0.996109, 0, 0D,
AxesLabel ▒ 8"t", "f"<D;
f
0.1
0.001
0.00001
1. ╣ 10-7
0
2
4
6
8
10
t
The double logarithmic plot of a relaxation process shows a shoulder and a
decay
pl1 = LogLogPlot@f@tD Й. sol2 Й. 8f@0D ▒ 1, W ▒ 1 Й 2<,
8t, .001, 10<, PlotStyle ▒ RGBColor@0.996109, 0, 0D,
AxesLabel ▒ 8"t", "f"<D;
f
0.1
0.001
0.00001
1. ╣ 10-7
0.001
0.01
0.1
1
10
t
7. Fractals
869
Both the log and the log-log plot show that a standard relaxation process
decays very fast. The decay of a b particle is a process determined by a
single physical cause. However, relaxation processes in complex materials
show a different characteristic pattern. The decay in complex materials is
much slower than in the standard relaxation case. The asymptotic behavior
observed can be described by a power law in time:
f HtL ≤ t-q , with 0 < q < 1.
(7.6.90)
The range of time extends over many decades. Examples are current
distributions at rough blocking electrodes [7.22], charge-carrier transport
in amorphous semiconductors [7.23], the dielectric relaxation of liquids
[7.24], and relaxation of polymeric networks [7.25?7.27].
One of the nonstandard relaxation models to describe the behavior of
complex materials assumes that the material has a memory. This memory
includes the total decay for all times. The model discussed by
Nonnenmacher [7.28] is applicable to models in which an integral net
effect determines the relaxation process. The relaxation equation is
generalized in such a way that a regular behavior at the initial time is
incorporated into the model. The equation is given by a Fredholm integral
equation of first kind expressed by RL differential operators. This kind of
relaxation process assumes that the order of differentiation is a positive
real number:
Assume@q, q > 0D;
The equation in terms of a RL operator reads
f0 tq
eq = +q0, t @f@tDD cccccccccccccccc
cccccccccccccccc + Wq f@tD == 0
Gamma@1 qD
f0 tq
Wq f@tD cccccccccccccccc
cccccc + +qt @f@tDD == 0
* @1 qD
The solution of this equation follows by applying FractalDSolve[] to the
fractional equation
870
7.6 Fractional Calculus
solf = -+tf @eqD
t q
9f ▒ FunctionAt, f0 Eq,1 A J cccc N EE=
W
The solution consists of a regular solution containing the initial condition
f0 . The generalized Mittag?Leffler function is nonstandard in
Mathematica. The graphical representation of the Mittag?Leffler function
for q = 1 Й 3 and f0 = 1 is given by
pl2 =
LogLogPlot@f@tD Й. solf Й. 8f0 ▒ 1, W ▒ 1 Й 2, q ▒ 1 Й 3<,
8t, .001, 10<, PlotStyle ▒ RGBColor@0.996109, 0, 1D,
AxesLabel ▒ 8"t", "f"<D;
f
1
0.7
0.5
0.3
0.001
0.01
0.1
1
10
t
Comparing the derived nonstandard relaxation result with the standard
relaxation solution demonstrates
7. Fractals
871
Show@pl1, pl2D;
f
0.1
0.001
0.00001
1. ╣ 10-7
0.001
0.01
0.1
1
10
t
that a fractional relaxation process decays much slower than a Debye
relaxation. This slower decay of a relaxation process is frequently
observed in natural systems.
7.6.4.6 Relaxation Oscillation Equation
Next, let us consider an equation which interpolates between the ordinary
relaxation and the oscillation equation. This kind of equation can be
considered as a weak form of Newton's equation or a generalization of
relaxation processes. The main assumption is that we restrict the order of
differentiation to the interval 1 ╖ q ╖ 2.
Assume@1 < q && q <= 2D
888q > 1, q ├ 2<, 8Im@qD ▒ 0, Re@qD ▒ q<<<
The equation under consideration is given by
872
7.6 Fractional Calculus
relaxOscill = +q0, t @f@tDD + f@tD == f@0D tqD
f@tD + +qt @f@tDD == tqD f@0D
where we specialized the left-hand side of the equation to a power
function. This equation is called the relaxation oscillation equation.
Applying the fractional solution operator to this equation will deliver the
solution
sol = -+tf @relaxOscillD
8f ▒ Function@t, tD f@0D * @1 q DD Eq,1D @ tq DD<
The result is a function determined by the generalized Mittag?Leffler
function Eq, p HtL providing us the solution manifold for different
differentiation orders q. Since the gamma function contained in this
solution possesses singularities at different negative integer orders of the
arguments, we have to choose the initial conditions in such a way that this
singularity is eliminated. We introduce a scaled initial condition
gH0L Й GH1 - q - aL, allowing us to exclude the singularity from the
functional domain. However, we must keep in mind that at certain values
of q = 1 - a, negative integers singularities of the function occur. The
following plot of the singularity free function shows the transition from
relaxation behavior to oscillations. Depending on the fractional order q, we
observe that the total relaxation phenomenon is converted to a damped
oscillation and then to an undamped oscillation if q increases from 1 to 2.
7. Fractals
873
Plot3D@Evaluate@
f@tD Й. sol Й. f@0D > g@0D Й * @1 q DD Й. D > 0.1 Й.
g@0D > 1D, 8t, 0.1, 12<, 8q, 1.0001, 2<,
AxesLabel > 8"t", "q", "f "<, PlotRange > All,
PlotPoints > 35, Mesh > FalseD;
1
0.5
f
0
-0.5
-1
2
1.8
1.6
1.4 q
2.5
5
t
1.2
7.5
10
The following contour plot of the solution shows that the frequency
decreases slightly if q is increased. However, this frequency shift
disappears for q values near 2.
874
7.6 Fractional Calculus
ContourPlot@Evaluate@
f@tD Й. sol Й. f@0D > g@0D Й * @1 q DD Й. D > 0.1 Й.
g@0D > 1D, 8t, 0.1, 12<, 8q, 1.0001, 2<, Axes > True,
AxesLabel > 8"t", "q"<, PlotRange > All,
ColorFunction > Hue, PlotPoints > 35D;
q
2
1.8
1.6
1.4
1.2
1
0
2
4
6
8
10
12
t
7.6.4.7 Semifractional Differential Equations
Semifractional differential equations are those equations with differential
order q = 1 Й 2. This kind of equation is in use in different fields of
chemistry and physics such as electroanalysis, polymer physics, and so
forth. A characteristic equation of relaxation type for a positive relaxation
time constant
7. Fractals
875
Assume@W > 0D
888W > 0<, 8Im@WD ▒ 0, Re@WD ▒ W<<<
is given by
f0 t1Й2
sfDG = +1Й2
cccccccccccccccc W1Й2 f@tD
0, t @f@tDD cccccccccccccccc
Gamma@1 Й 2D
1
f0
f@tD
ccc
2c
cccccccc
Х!!!!cccccccc
Х!!!!ccc cccccccc
Х!!!!ccccc + +t @f@tDD
S t
W
The solution of this equation is derived by applying the fractional solution
operator -+tf to the fractional differential equation:
sDGSol = -+tf @sfDGD
i
j
1 Х!!!!
j1 H1 + фtЙW L $%%%%%%
9f ▒ FunctionAt, f0 j
cccc % W +
j
j
W
k
z
1 Х!!!!
1 y
Х!!!!
zE=
фtЙW $%%%%%%
cccc % W ErfA t $%%%%%%
cccc % Ez
z
W
W z
{
The result is a function combining exponentials and error functions. A plot
of the solution is given next for different relaxation constants t.
876
7.6 Fractional Calculus
Plot3D@f@tD Й. sDGSol Й. 8f0 > 1<, 8t, 0, 4<,
8W, 0.2, 1<, AxesLabel > 8"t", " W", "f "<,
Mesh > False, PlotPoints > 35D;
1.8
f 1.6
1.4
1
0.8
0.6 t
0
1
0.4
2
t
3
4
0.2
Another example for a semifractional equation is given by the driven
rubber equation:
drfDG = )@ t D + Wb +1Й2
0, t @)@tDD a0 Sin@Z tD == 0
1
ccc
c
Wb +t 2 @)@tDD Sin@t ZD a0 + )@tD == 0
This kind of equation is used to model the relaxation behavior of rubber
driven by a harmonic external force. The solution of the equation is gained
by application of the fractional solution operator:
7. Fractals
877
solution = -+t) @drfDGD
1
9) ▒ FunctionAt, cccccccccccccccc
cccccccccccccccccc
4 H1 + W4 b Z2 L
81<
i
i Х!!!!!!!! Х!!!!!!!!
1
j
j
j
; ccc
j
c t2 Z2 E
2 2 S t Z Fp,q A 3
j
5
j
4
j
j
8
ccc
c , ccc
c<
j
j
Х!!!!
b
4
4
j
j
j
Z j
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccccccccccccccc +
W
j
j
5
j
j
j
j
* @ ccc34c D * @ ccc
cD
j
j
4
j
j
j
j
k
k
Х!!!!
Wb Z
i
j
j
Х!!!!!!!!!!!!
j
j
j
4 S W2 b t2 Z2
j
j
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccc
cccccccc Х!!!!!!!!!!!!!!!
Х!!!!!!!!!!!!!!!
j
j
t2 Z2
t2 Z2
j
cccccc E * A1 cccccccc
cccccc E
j t * A cccccccc
j
S
S
j
k
4S
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccccccccccc
ccccccccc Х!!!!!!!!!!!!!!!
Х!!!!!!!!!!!!!!!
t2 Z2
t2 Z2
1
1
* A ccc
c cccccccc
cccccc E * A ccc
c + cccccccc
cccccc E
2
S
2
S
81<
Х!!!!!!!! b Х!!!!
1
c t2 Z2 E
Z Ht ZL3Й2 Fp,q A 5
; ccc
2S W
7
4
c , ccc
c<
8 ccc
4
4
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccc
7
* @ ccc54c D * @ ccc
cD
4
1
c <<
881, ccc
y
Х!!!!!!!!!!!!!!! yy
z
2
z
z
z
4 <1,0 A
; 2 t W2 b E z
z
z
z
z
z
z
z
8<
z
z
z
z
z
z
z
z
z
cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccc
c
cccccccc
c
E=
a
0
z
z
z
Х!!!! Х!!!!!!!!!!!!!!!
z
z
z
2
b
z
z
z
S tW
z
z
zz
z
z z
z
z
{{ {
The result is a combination of generalized hypergeometric functions. This
solution demonstrates that a more or less simple initial equation results in a
complicated structure of the solution. An example of the solution is shown
in the following plot. The parameters used are t = 1, b = 1 Й 3, a0 = 1, and
w = 1.4.
878
7.6 Fractional Calculus
Plot@Evaluate@)@ t D Й. solution Й.
8W > 1, b > 1 Й 3, a0 > 1, Z > 1.4<D,
8t, 0.1, 15<, AxesLabel > 8"t", "f"<D;
f
0.4
0.2
-0.2
2
4
6
8
10
12
14
t
-0.4
-0.6
7.6.4.8 Anomalous Diffusion
Many experiments indicate that diffusion processes usually do not follow
the standard Gaussian behavior. In turn, the mean square displacement
XrHtL2 \ ≤ t for a Gaussian process changes to XrHtL2 \ ≤ t2Йdw , where the
anomalous diffusion exponent dw differs from 2, the value for standard
(Fickean) diffusion. The deviation from a linear dependence to a power
law is an indication for anomalous diffusion. Anomalous diffusion in
which the mean square distance between diffusing quantities increases
slower or faster than linearly in time has been observed in different
physical and biological systems from macroscopic surface growth to DNA
sequences [7.29]. One of the first investigations discussing fractional
diffusion goes back to Wyss [7.30] and O'Shaugnessy and Procaccia
[7.31]. A method for solving fractional diffusion equations using Fox's H
functions has been presented by Schneider and Wyss [7.32] and more
recently by Metzler et al. [7.33].
The motivation for the anomalous diffusion equation being discussed now
follows the ideas already outlined in the section on fractional relaxation
starting from the standard model and generalizing the equation by
7. Fractals
879
incorporating the initial condition. The standard (Fickean) diffusion
equation in 1+1-dimensions reads
≥t rHx, tL = ≥x,x rHx, tL,
(7.6.91)
with an additional initial condition rHx, t = 0L = r0 HxL. Equation (7.6.91) is
given in a scaled representation where the diffusion constant is
incorporated as a factor in the time variable. Let us start with the
memory-diffusion equation
t
≥t rHx, tL = ?0 KHt - tL ≥x,x rHx, tL ? t,
(7.6.92)
which has already been motivated and derived recently [7.34, 7.35].
Again, as in the case of relaxation, we assume that the memory kernel
takes on a power law KHtL = D0 t b-1 Й GHbL with 0 < b < 1. Then we can
express Eq. (7.6.92) by
D
t
0
ее Ht - tL b-1 ≥x,x rHx, tL ? t and b > 0,
≥t r = ееееееее
GH bL ?0
(7.6.93)
which, in terms of Riemann?Liouville operators +a0, t , can be written as
-b
+10, t rHx, tL = D0 +0, t H≥x,x rHx, tLL.
(7.6.94)
Applying an integration +-1
0, t to both sides of Eq. (7.6.94), we find
-H1+ bL
rHx, tL - r0 HxL = D0 +0, t
H≥x,x rHx, tLL.
(7.6.95)
A differentiation of order H1 + bL of Eq. (7.6.95) replaces the integral
representation of the generalized diffusion equation by its differential
representation
1+ b
t-H1+bL
еееее = D0 ≥x,x rHx, tL.
+0, t rHx, tL - r0 HxL ееееееее
GH- bL
(7.6.96)
This generalized diffusion equation incorporates, in addition to the
fractional differentiation in time, the initial condition r0 for the density r.
Replacing the fractional order 1 + b by q, we find the simplified equation
q
t-q
ееееееее = D0 ≥x,x rHx, tL with 1 < q < 2. (7.6.97)
+0, t rHx, tL - r0 HxL ееееееее
GH1-qL
The solution of Eq. (7.6.97) follows by the following steps. First, let us
assume
Assume@q > 0D;
880
7.6 Fractional Calculus
Next, define the equation
equation18 =
tq
+q0, t @U@x, tDD U0 @xD cccccccccccccccc
ccccc == D0 ≥x,x U@x, tD
*@1 qD
tq U0 @xD
+qt @U@x, tDD cccccccccccccccc
cccccccc == D0 UH2,0L @x, tD
* @1 qD
Then, apply the Laplace transform to Eq. (7.6.97):
equation18Laplace = 3st @equation18D
Conditions to solve the integral:
1 + Re@qD < 0
sq 3st @U@x, tDD s1+q U0 @xD == 3st @UH2,0L @x, tDD D0
The second step of the transformation consists of a Fourier transform of
the equation in Laplace space:
foulapgl2 = -xk @equation18Laplace Й.
83st @U@x, tDD ▒ L@xD, 3st @≥x,x U@x, tDD ▒ ≥x,x L@xD,
U0 @xD ▒ DiracDelta@xD, C1@xD ▒ 0<D
2 s1+q + sq -xk @L@xDD == k2 -xk @L@xDD D0
The algebraic solution in Fourier and Laplace space follows by
foulaploes2 = Solve@foulapgl2, -xk @L@xDDD ЙЙ Flatten
2 s1+q
9-xk @L@xDD ▒ cccccccccccccccc
ccccccc =
q
s + k2 D0
7. Fractals
881
The application of the inverse Fourier transform on this solution delivers
the solution in spatial and Laplacian variables Hx, sL:
laploes2 =
x
Map@H- 1 Lk @#D &, foulaploes2, 82<D Й. L@xD ▒ 3st @U@x, tDD
sq
H?x?L $%%%%%%%%%%
cccc
D0cc
Х!!!!!!!!
ф
s1+q sq
s
93t @U@x, tDD ▒ cccccccccccccccccccccccccccccccc
cccccc =
Х!!!!!!cccccccccccccccc
D0
The result shows that the Laplace solution contains a stretched exponential
multiplied by a power function. If we restrict our consideration to the
half-space x > 0 and assume that the diffusion constant D0 is positive,
Assume@x > 0, C1 > 0D;
we can represent the result in Mellin space as
mellaploes2 =
4zs @laploes2 Й. D0 ▒ C1 ЙЙ PowerExpandD ЙЙ PowerExpand ЙЙ
Simplify
1+z
2+q+2 z
2+q+2 z
q cc x cccccccccqccccccccc * A cccccccc
2 C1 cccccccc
ccccccccc E
q
9* @zD 41z
@U@x,
tDD
▒
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccccccccccc
ccccc =
t
q
A shift of the Mellin variable by 1 gives us the Mellin solution
melloes2 = Solve@mellaploes2 Й. 8z ▒ 1 z, Rule ▒ Equal<,
4zt @U@x, tDDD ЙЙ Flatten
z
94zt @U@x,
2+q+2 H1zL
2+q+2 H1zL
q cccccccccccc * A cccccccccccccccc
2 C1 cccqc x cccccccccccccccc
ccccccccc E
q
tDD ▒ cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
ccccccccccccccccc =
q * @1 zD
The inversion of the Mellin transform provides the final result:
882
7.6 Fractional Calculus
solution =
t
H41 Lz @melloes2D Й. C1 ▒ D0 ЙЙ PowerExpand ЙЙ Simplify
?
8< ╩
881, 1<<
?
?
?
?
E
2
?
?
ccc
c
==
?
8<
991,
?
q
?
9U@x, tD ▒ cccccccccccccccccccccccccccccccc
cccccccccccccccccccccccccccccccc
cccccccccccccccc
cccccccccccccccc
cccccccc =
qx
2
/
1,0
x2Йq D1Йq
0
1,1 A ccccccccctccccccccc
The solution of the generalized diffusion equation (7.6.97) thus is
1,0
represented by a Fox's H function of
series representation as follows:
?
?
1,0 x
?
?
/1,1
A cccccccctDccccccccc ?
?
?
?
2Йq
1Йq
0
┬
?
k=0
/1,1 . This function is given by a
8< ╩
881, 1<<
2
q << ╩
? 881, ccc
q H1Lk
cccccccccccccccc
cccccccccc
2 * H1 cccq2c cccccccc
H1+kLL
k!
8<
ccccc
i
y
x q D1Йq
0
j cccccccc
z
tccccccc
k
{
2
E=
(7.6.98)
cccq2c H1+kL
.
A graphical representation of the solution is given in Figure 7.6.27.
0.6
0.4
0.2r
5
0
4
3
t
1
2
2
x
1
3
Figure 7.6.27.
Solution of the fractional diffusion equation (7.6.97) in the series representation (7.6.98).
The fractional exponent is q = 3 Й 2 and D0 = 1.
7. Fractals
883
7.7 Exercises
1. Use the Tree[] function to create different kinds of trees. Which
option determines the shape of a tree?
2. Extend the Koch[] function to other generators (e.g., the Peano
curve). For a set of generators consult the book by Mandelbrot [7.4].
3. Examine the multifractal properties of a system with different
numbers of probabilities and scaling factors. Determine the fractal
dimensions D0 and D1 and give a graphical representation of these
dimensions versus the number of scaling factors.
4. Use hexagonal lattices in the renormalization procedure for the
percolation model.
7.8 Packages and Programs
7.8.1 Tree Generation
This package allows one to generate a fractal tree.
BeginPackage["Tree`"];
Needs["Geometry`Rotations`"];
Clear[Tree, rotateLine, branchLine, createBranches];
Tree::usage = "Tree[options___] creates a fractal
tree. The options of
the function Tree determine the form of the fractal
created. Options are
Generation -> 10, \n
BranchRotation -> 0.65, \n
BranchScaling -> 0.75, \n
BranchThickness -> 0.7, \n
OriginalThickness -> 0.07, \n
BranchColor -> {RGBColor[0,0,0]} \n
Example: Tree[BranchColor->l1,BranchRotation->0.3] \n
884
7.8 Packages and Programs
l1 is a list created in the package Tree.";
(* --- global variables --- *)
Generation::usage;
BranchRotation::usage;
BranchSkaling::usage;
BranchThickness::usage;
OriginalThickness::usage;
BranchColorn::usage;
Begin["`Private`"];
(* --- rotate a line --- *)
rotateLine[Line[{start_, end_}], angle_] :=
Line[{end, end + branchScaling*
Rotate2D[end - start, angle Random[Real,
{0.5,1.5}]
]}];
(* --- branch a line --- *)
branchLine[Line[points_]] :=
{rotateLine[Line[points],
branchRotation],
rotateLine[Line[points],
- branchRotation]};
(* --- change thickness --- *)
branchLine[Thickness[th_]] := Thickness[th
branchThickness];
(* --- define color of a branch --- *)
branchLine[RGBColor[r_, g_, b_]] := (
branchColor = RotateLeft[branchColor];
First[branchColor] );
(* --- create branches --- *)
createBranches[lines_] := Flatten[Map[branchLine,
lines]];
(* --- options if Tree[] --- *)
Options[Tree] =
{
Generation -> 10,
BranchRotation -> 0.65,
BranchScaling -> 0.75,
BranchThickness -> 0.7,
OriginalThickness -> 0.07,
7. Fractals
885
BranchColor ->
{
RGBColor[0,0,0]}
};
(* --- create a tree --- *)
Tree[options___] := Block[
{generations, branchRotation,
branchScaling, branchThickness,
originalThickness, branchColor},
(* --- check options --- *)
{generations, branchRotation, branchScaling,
branchThickness,
originalThickness, branchColor} =
{Generation, BranchRotation, BranchScaling,
BranchThickness,
OriginalThickness, BranchColor} /. {options}
/. Options[Tree];
(* --- iterate the functions and display the tree
--- *)
Show[
Graphics[
NestList[
createBranches,
{
First[branchColor],
Thickness[originalThickness],
Line[{{0,0},{0,1}}]
},
generations]],
FilterOptions[Show, options],
AspectRatio -> Automatic,
PlotRange -> All]];
(* --- filter for options --- *)
FilterOptions[ command_Symbol, opts___] :=
Block[{keywords = First /@ Options[command]},
Sequence @@ Select [{opts},
MemberQ[keywords, First[#]]&]];
End[];
EndPackage[];
(* --- an example of a color list --- *)
l1 = {RGBColor[0.5620000000000001, 0.236, 0.071],
886
7.8 Packages and Programs
RGBColor[0.5470000000000001, 0.229,
0.06900000000000001],
RGBColor[0.5, 0.21, 0.063], RGBColor[0.469,
0.196, 0.059],
RGBColor[0.033, 0.281, 0.035], RGBColor[0.046,
0.395, 0.05],
RGBColor[0.055, 0.469, 0.059],
RGBColor[0.07000000000000001, 0.602, 0.076],
RGBColor[0.085, 0.727, 0.092], RGBColor[0.109,
0.937, 0.118],
RGBColor[0.013, 0.75, 0.028]};
7.8.2 Koch Curves
This package generates fractal curves of a different kind.
BeginPackage["Koch`"];
Clear[Koch,VKoch,WKoch,QKoch,Quad,NGon,docurve,Fracta
l,FaktalPlot];
Needs["Geometry`Rotations`"];
Fractal::usage = "Fractal[curve_String, options___]
creates a graphical
representation of a fractal curve. The type of curve
is determined by
the first argument. A list of available curves is
obtained by calling
Fractal[List] or Fractal[Help]. The second argument
allows to change the
options of the function. The default values are
Generations -> 3,
Angle -> Pi/6 and Corners -> 6.";
Generations::usage;
Angle::usage;
Corners::usage;
Begin["`Private`"];
(* --- generator of the Koch curve --- *)
(*
__/\__
*)
Koch[Line[{StartingPoint_,EndPoint_}]]:=Block[{fActor
7. Fractals
887
, angle, liste={}},
fActor = 1/3;
angle = Pi/3;
l1 = StartingPoint;
l2 = StartingPoint+(EndPoint StartingPoint)*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,-angle,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,angle,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,0,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]]
];
(* --- generator of an altered Koch curve --- *)
(*
____/\
*)
VKoch[Line[{StartingPoint_,EndPoint_}]]:=Block[{fActo
r, angle, liste={}},
fActor = 1/3;
angle = Pi/3;
l1 = StartingPoint;
l2 = StartingPoint+(EndPoint StartingPoint)*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,0,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,-angle,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,angle,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]]
];
(* --- generator of the Koch curve with variable
base angle --- *)
888
7.8 Packages and Programs
WKoch[Line[{StartingPoint_,EndPoint_}]]:=Block[{fActo
r, liste={}},
fActor = 1/(2*(1+Cos[angle]));
l1 = StartingPoint;
l2 = StartingPoint+(EndPoint StartingPoint)*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,-angle,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,angle,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,0,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]]
];
(* --- generator of the rectangular Koch curve --- *)
(*
__
__| |
__
|__|
*)
QKoch[Line[{StartingPoint_,EndPoint_}]]:=Block[{fActo
r, angle, liste={}},
fActor = 1/4;
angle = Pi/2;
l1 = StartingPoint;
l2 = StartingPoint+(EndPoint StartingPoint)*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,-angle,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,0,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,angle,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
7. Fractals
889
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,angle,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,0,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,-angle,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,0,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]]
];
(* --- generator for a rectangular curve --- *)
(*
__
__| |__
*)
Quad[Line[{StartingPoint_,EndPoint_}]]:=Block[{fActor
, angle, liste={}},
fActor = 1/3;
angle = Pi/2;
l1 = StartingPoint;
l2 = StartingPoint+(EndPoint StartingPoint)*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,-angle,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,0,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,angle,{0,0}]*fActor;
AppendTo[liste,Line[{l1,l2}]];
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,0,{0,0}]*fActor;
(* l2 = l2 + EndPoint*fActor;*)
AppendTo[liste,Line[{l1,l2}]]
];
890
7.8 Packages and Programs
(* --- generator for a N gon --- *)
NGon[Line[{StartingPoint_,EndPoint_}]]:=Block[{l1,
l2, angle, liste={}},
angle = 2*Pi/corners;
l1 = StartingPoint;
l2 = StartingPoint+(EndPoint - StartingPoint);
AppendTo[liste,Line[{l1,l2}]];
Do[
l1 = l2;
l2 = l2 +
Rotate2D[EndPoint-StartingPoint,k*angle,{0,0}];
AppendTo[liste,Line[{l1,l2}]],
{k,1,corners-1}];
liste
];
(* --- calculate the higher iterations --- *)
docurve[Type_,Linie_]:=Block[{},
Flatten[Map[Type,Linie]]
];
(* --- plot of a line sequence --- *)
FractalPlot[x_]:=Show[Graphics[x],AspectRatio->Automa
tic];
(* --- options for Fractal[] --- *)
Options[Fractal]:={
Generations -> 3,
Angle -> Pi/6,
Corners -> 6
};
(* --- create the fractal curve --- *)
Fractal[curve_, options___]:=Block[{generations,
angle, corners},
(* --- check options --- *)
{generations,angle,corners} =
{Generations,Angle,Corners}
/. {options} /. Options[Fractal];
(* --- menu for the different fractal curves --- *)
If[curve == "List" || curve == "Help",
Print[" "];
Print[" --------- available curves ---------"];
7. Fractals
891
Print["
Print["
Print["
Print["
Print["
Print["
Print["
Print["
Print["
Print["
Koch
:
QKoch :
VKoch :
WKoch :
Quad
:
Star
:
Square :
N-gon :
Mixture:
Random :
Koch curve"];
rectangular Koch curve"];
altered Koch curve"];
variable angle Koch curve"];
rectangular curve"];
Koch star"];
Koch square"];
Koch N gon"];
2 x Koch and 2 x QKoch"];
random generation"]];
(* --- plot the Koch curves --- *)
If[curve == "Koch" ||
curve == "QKoch" ||
curve == "VKoch" ||
curve == "WKoch" ||
curve == "Quad",
(* --- ToExpression transforms a string to an
expression --- *)
k1 =
ToExpression[curve][Line[{{0,0},{1,0}}]];
Do[
k1 = docurve[ToExpression[curve],k1],
{k,1,generations}];
FractalPlot[k1]
];
(* --- plot a Koch star --- *)
If[curve == "Star",
corners = 3;
k1 = NGon[Line[{{0,0},{1,0}}]];
Do[
k1 = docurve[Koch,k1],
{k,1,generations}];
FractalPlot[k1]
];
(* --- plot a Koch square --- *)
If[curve == "Square",
corners = 4;
k1 = NGon[Line[{{0,0},{1,0}}]];
Do[
k1 = docurve[Koch,k1],
{k,1,generations}];
FractalPlot[k1]
];
(* --- plot a Koch N gon --- *)
If[curve == "N-gon",
k1 = NGon[Line[{{0,0},{1,0}}]];
Do[
892
7.8 Packages and Programs
k1 = docurve[Koch,k1],
{k,1,generations}];
FractalPlot[k1]
];
(* --- plot a mixture of Koch curves --- *)
If[curve == "Mixture",
k1 = Koch[Line[{{0,0},{1,0}}]];
k1 = docurve[Koch,k1];
k1 = docurve[QKoch,k1];
k1 = docurve[QKoch,k1];
FractalPlot[k1]
];
(* --- plot a random sequence of Koch curves --- *)
If[curve == "Random",
listec ={Koch,QKoch,VKoch,WKoch,Quad,NGon};
k2 = Random[Integer,{1,6}];
k3 = Random[Integer,{1,6}];
If[k2 == 6 || k3 == 6, corners =
Random[Integer,{3,12}]];
name1 = listec[[k2]];
name2 = listec[[k3]];
k1 = name1[Line[{{0,0},{1,0}}]];
k1 = docurve[name1,k1];
Do[
k1 = docurve[name2,k1],
{k,1,generations-1}];
FractalPlot[k1]
];
];
End[];
EndPackage[];
7.8.3 Multifactals
The multifractal package provides functions for the determination of
multifractal spectra.
BeginPackage@"MultiFractal`"D;
Clear@Dq, Tau, Alpha, MultiFractalD;
MultiFractal::usage =
"MultiFractal@p_List,r_ListD calculates the
multifractal spectrum D_q for a model
7. Fractals
893
based on the probabilities p and the
scaling factors r. This function plots
five functions TauHqL, D_qHqL,
AlphaHqL, fHqL and fHAlphaL.";
Begin@"`Private`"D;
Hcalculate the multifractal dimensionsL
Dq@p_List, r_ListD := Block@8l1, l2, listrg = 8<<,
Hlength of the listsLl1 = Length@pD;
l2 = Length@rD;
If@l1 m l2,
Hvariation of q and determination of D_qL
Do@gl1 = Sum@p@@jDD ^ q r@@jDD ^ HHq 1L DfractalL,
8j, 1, l1<D 1;
result = FindRoot@gl1 m 0, 8Dfractal, 3, 3<D;
result = Dfractal Й. result;
Hcollect the results in a listL
AppendTo@listrg, 8q, result<D,
8q, 10, 10, .101<D, Print@" "D;
Print@" Lengths of lists are different!"D;
listrg = 8<D;
listrgD;
Hcalculate TauL
Tau@result_ListD := Block@8l1, listtau = 8<<,
Hlengths of the listsL
l1 = Length@resultD;
Hcalculate TauLDo@
AppendTo@listtau, 8result@@k, 1DD, result@@k, 2DD
H1 result@@k, 1DDL<D, 8k, 1, l1<D;
listtauD;
HLegendre transformL
Alpha@result_ListD := Block@8l1, dq, listalpha = 8<,
listf = 8<, listleg = 8<, mlist = 8<, pl1, pl2<,
Hlengths of the listsL
l1 = Length@resultD;
Hdetermine the differential dqL
894
7.8 Packages and Programs
dq = Hresult@@2, 1DD result@@1, 1DDL 2;
Hcalculate Alpha by numerical
differentiationLDo@
AppendTo@listalpha, 8result@@k, 1DD,
Hresult@@k + 1, 2DD result@@k 1, 2DDL Й
dq<D, 8k, 2, l1 1<D;
l2 = Length@listalphaD;
Hcalculate f and
collect the results in a listLDo@
AppendTo@listf, 8result@@k, 1DD, Hresult@@k, 1DD
listalpha@@k, 2DD result@@k, 2DDL<D;
listalpha@@k, 2DD = listalpha@@k, 2DD,
8k, 1, l2<D;
Hlist of the Legendre transformsL
Do@AppendTo@listleg,
8listalpha@@k, 2DD, listf@@k, 2DD<D;
AppendTo@mlist, listf@@k, 2DDD, 8k, 1, l2<D;
Hplot f and alpha versus qLpl1 =
ListPlot@listalpha, PlotJoined ▒ True, AxesLabel ▒
8"q", "D"<, Prolog ▒ Thickness@0.001DD;
pl2 = ListPlot@listf, PlotJoined ▒ True, AxesLabel ▒
8"q", "f"<, Prolog ▒ Thickness@0.001DD;
Show@8pl1, pl2<, AxesLabel ▒ 8"q", "D,f"<D;
Hplot the Legendre
transform f versus alphaL
ListPlot@listleg, AxesLabel ▒ 8"D", "f"<D;
Hprint the maximum of f=D_ 0L
maxi = Max@mlistD;
Print@" "D;
Print@"
D_0 = ", maxiDD;
Hcalculate the multifractal propertiesL
MultiFractal@p_List, r_ListD :=
Block@8listDq, listTau<,
Hdetermine D_qLlistDq = Dq@p, rD;
ListPlot@listDq, PlotJoined ▒ True, AxesLabel ▒
8"q", "Dq "<, Prolog ▒ Thickness@0.001DD;
Hcalculate TauLlistTau = Tau@listDqD;
ListPlot@listTau, PlotJoined ▒ True, AxesLabel ▒
8"q", "W"<, Prolog ▒ Thickness@0.001DD;
Hdetermine the Hoelder exponentL
7. Fractals
895
Alpha@listTauDD;
End@D;
EndPackage@D;
7.8.4 Renormalization
This package supports the calculations of renormalization.
BeginPackage["Renormalization`"];
Clear[f, Pcrit, Nc, Dim, RenormPlot];
Nc::usage = "Nc[n_] determines the mean number of
atoms at the probability
p_c if m is changed in the range 1 <= m <= n-2. The
size of the block is
determined by n.";
Dim::usage = "Dim[n_] calculates the fractal
dimension for the critical
probability p_c. The dimension depends on m where 1
<= m <= n-2, n is
the size of the block used.";
Pcrit::usage = "Pcrit[n_] determines the critical
probability for an n x n
grid under the variation of m where m is the number
of empty locations in the
grid. The range of m is 1 <= m <= n-2.";
RenormPlot::usage = "RenormPlot[n_,typ_String] plots
the functions Nc, Dim
or Pcrit.";
Begin["`Private`"];
(* --- auxilary function --- *)
f[p_,n_,m_]:=
Sum[Binomial[n,i]*p^(n-i)*(1-p)^i,{i,0,m}];
(* --- mean number of particles on a grid --- *)
Nc[n_]:= Block[{p, ncliste={}},
p = Pcrit[n];
896
7.8 Packages and Programs
Do[
AppendTo[ncliste,
Sum[Binomial[n,i]*(n-i)*p[[k]]^(n-i-1)*(1-p[[k]])^i,
{i,0,k}]],
{k,1,n-2}];
ncliste
];
(* --- fractal dimension at the critical probability
--- *)
Dim[n_]:=
N[Log[Nc[n]]/Log[Sqrt[n]]];
(* --- critical probability on a n x n grid --- *)
Pcrit[n_]:=Block[{ph, p, erg, erg1, gl1, pliste1={}},
If[n > 2,
Do[
gl1 = p - f[p,n,i];
(* --- solution of the fixpoint equation --- *)
erg = NSolve[gl1==0,p];
erg = p /. erg;
(* --- use only real solutions --- *)
erg1 = {};
Do[If[Head[erg[[k]]]==Real,AppendTo[erg1,erg[[k]]]],
{k,1,Length[erg]}];
(* --- looking for solutions between 0 and 1
--- *)
erg = Sort[erg1];
erg1 = {};
Do[If[erg[[k]] > 0.0 ,
AppendTo[erg1, erg[[k]] ] ],
{k,1,Length[erg]}];
ph = Min[erg1];
AppendTo[pliste1,ph],
{i,1,n-2}],
Print["
"];
Print[" choose n > 2 "]];
pliste1];
(* --- plot the results --- *)
RenormPlot[n_,typ_String]:=Block[{},
If[typ == "Pcrit",
ListPlot[Pcrit[n],AxesLabel->{"m","pc"}],
If[typ == "Nc",
7. Fractals
897
ListPlot[Nc[n],AxesLabel->{"m","Nc"}],
If[typ == "Dim",
ListPlot[Dim[n],AxesLabel->{"m","D"}],
Print[" "];
Print[" Wrong key word use: "];
Print[" Pcrit, Nc or Dim.
"];
Print[" "]
]
]
]
];
End[];
EndPackage[];
7.8.5 Fractional Calculus
Define the global variable $FractionalCalculusPath in such a way that the
location of the package FractionalCalculus is uniquely defined.
$FractionalCalculusPath =
$AddOnsDirectory <> "ЙApplicationsЙFracCalcЙ";
AppendTo@$Path, $FractionalCalculusPathD;
Load the package:
<< FractionalCalculus.m
--> FractionalCalculus ready <-╘ Gerd Baumann, Norbert SЭdland 1996-2004
<< Integral.m
-- "Integral.m" is available. --
NotebookClose@foxtitleD;
A
Appendix
This appendix contains some information on the installation of the
accompanying software and a short description of the functions defined in
the packages. It also summarizes the Mathematica functions used in the
book.
A.1 Program Installation
The book is accompanied by a CD containing all Mathematica notebooks.
These notebooks can be used as interactive text. The notebooks are linked
to a style file called ScriptStyle.nb and Vortrag.nb. You should copy these
two files to the location where the additional style files are stored. For
example,
on
a
PC,
the
style
files
are
located
at
C:\WINDOWS\Profiles\All_Users\Applications\Mathematica\SystemFiles\
FrontEnd\StyleSheets.
In addition to the notebooks, there is the package EulerLagrange which is
delivered with the text. The package is used in Chapter 2. In this chapter
you have to change the path name in the sections Packages and Programs.
900
1.1 Program Installation
You can either use the package from the CD or you can copy the package
to your preferred location. In any case, you have to change the path name
of the package.
Other packages supporting calculations of the text are located in the
section Packages and Programs in the appropriate notebook. For these
packages, there is no need to set any path names. They are ready to use for
your calculations.
A.2 Glossary of Files and Functions
This section contains a short description of all functions defined in the
packages of this book. The packages are alphabetically listed.
A.2.1 Anharmonic Oscillator
Anharmonic oscillator of quantum mechanics.
Х
AsymptoticPT
AsymptoticPT[N_,kin_] determines the asymptotic approximation for
╩ x |ь╤ for the continuous case of eigenvalues in a PЖschel?Teller
potential. The function yields an analytical expression for ╩ bHkL ╩2 . The
variables Transmission and Reflection contain the expressions for the
transmission and the reflection coefficients. w1a and w2a contain the
approximations for x ь -╤ and x ь ╤, respectively.
Х
PoeschelTeller
PoeschelTeller[x_, n_, N_] calculates the eigenfunction of the
PЖschel?Teller potential for discrete eigenvalues. N determines the depth
of the potential V0 SechHxL by V0 = NHN + 1L. n fixes the state where
0 < n ╖ N.
Х PlotPT
A. Appendix
901
PlotPT[kini_,kend_,type_] gives a graphical representation of the
reflection or transmission coefficient depending on the value of the
variable type. If type is set to the string r, the reflection coefficient is
plotted. If type is set to t, the transmission coefficient is represented. This
function creates five different curves.
Х Reflection
Variable containing the reflection coefficient. The independent variables
are N and k.
Х Transmission
Variable containing the expression for the transmission coefficient. The
independent variables are N and k.
Х w1a
The variable contains the analytic expression for the asymptotic
approximation for x ь -╤.
Х w2a
The variable contains the analytic expression for the asymptotic
approximation for x ь ╤.
A.2.2 Boundary Value Problem of Electrodynamics
Boundary value problem of electrodynamics.
Х Potential
Potential[boundary_,R_,alpha_,n_] calculates the potential in a circular
segment. Input parameters are the potential on the circle, the radius R of
the circle, and the angle of the segment of the circle. The last argument n
determines the number of expansion terms used to represent the solution.
902
1.2 Glossary
A.2.3 Central Field Problem in Quantum Mechanics
Quantum mechanical description of motion in a central field.
Х Angle
Angle[theta_, phi_, l_, m_] calculates the angular part of the wave function
for an electron in the Coulomb potential. The numbers L and m denote the
quantum numbers for the angular momentum operator. q and f are the
angles in the spherical coordinate system.
Х AnglePlot
AnglePlot[pl_,theta_,phi_] gives a graphical representation of the function
contained in pl. The range of representation is p ╖ f < 5 p Й 2 and 0 < q < p
. q is measured with respect to the vertical axis. This function is useful for
plotting the orbitals and the angular part of the eigenfunction.
Х Orbital
Orbital[theta_,phi_,l_,m_,type_String] calculates the superposition of two
wave functions for the quantum numbers ml = +m and ml = -m. The
variable type allows the creation of the sum or the difference of the wave
functions. The string values of type are either plus or minus.
Х Radial
Radial[ro_, n_, l_, Z_] calculates the radial representation of the
eigenfunctions for an electron in the Coulomb potential. The numbers n
and l are the quantum numbers for the energy and the angular momentum
operator. Z specifies the number of charges in the nucleus. The radial
distance between the center and the electron is given by r.
A.2.4 Harmonic Oscillator in Quantum Mechanics
Х a
A. Appendix
903
a[psi_, xi_:x] is annihilation operator for eigenfunction y. The second
argument specifies the independent variable of the function y.
Х across
across[psi_, xi_:x] is creation operator for eigenfunction y. The second
argument specifies the independent variable of y.
Х Psi
Psi[xi_,n_] represents the eigenfunction of the harmonic oscillator. The
first argument x is the spatial coordinate. The second argument n fixes the
eigenstate.
Х wcl
wcl[xi_,n_] calculates the classical probability of locating the particle in
the harmonic potential. The first argument x is the spatial coordinate and n
determines the energy given as the eigenvalue.
Х wqm
wqm[xi_,n_] calculates the quantum mechanical probability for an
eigenvalue state n. The first argument represents the spatial coordinate.
?
A.2.5 Korteweg?de Vries Equation
Multisoliton solution of the Korteweg?de Vries equation.
Х Soliton
Soliton[x_,t_,N_] creates the N soliton solution of the KdV equation.
Х PlotKdV
PlotKdV[tmin_,tmax_,dt_,N_] calculates a sequence of pictures for the N
soliton solution of the KdV equation. The time interval of the
904
1.2 Glossary
representation is @tmin , tmax D. The variable dt measures the length of the
time step.
A.2.6 Korteweg de Vries equation and its derivation
Derivation of the Korteweg de Vries equation.
Х Equation
Equation[n_] calculates the evolution equation up to order n.
A.2.7 Korteweg?de Vries Equation and Integrals of Motion
Integral of motion of the Korteweg?de Vries equation.
Х Gardner
Gardner[N_] calculates the densities of the integrals of motion for the KdV
equation using Gardner's method. The integrals are determined up to the
order N.
A.2.8 Korteweg?de Vries Equation Numerical Solution
Numerical solution of the Korteweg?de Vries equation.
Х KdVNIntegrate
KdVNIntegrate[initial_,dx_,dt_,M_] carries out a numerical integration of
the KdV equation using the procedure of [3.5]. The input parameter
initially determines the initial solution in the procedure (e.g., -6 Sech2 HxL).
The infinitesimals dx and dt are the steps with respect to the spatial and
temporal directions. M fixes the number of steps along the x-axis.
A. Appendix
905
A.2.9 Koch Curves
Fractal curves.
Х Fractal
Fractal[curve_String, options___] creates a graphical representation of a
fractal curve. The type of curve is determined by the first argument. A list
of available curves is obtained by calling Fractal[List] or Fractal[Help].
The second argument allows changing the options of the function. The
default values are Generations ь 3, Angle ь p Й 6 and Corners ь 6.
A.2.10 Light Beam Near a Planet
The bending of a light beam near a planet is discussed.
Х Deviation
Deviation[radius_,mass_] calculates the numerical value of the light
bending in a gravitational field of a star with mass M in a distance radius
of the center.
Х Orbit
Orbit[radius_,mass_] plots the orbit of a light beam near a mass in the
distance radius. The calculation is done in Schwarzschild metric.
A.2.11 Multifractal Properties
Multifractal properties of point sets.
Х MultiFractal
MultiFractal[p_List,r_List] calculates the multi-fractal spectrum Dq for a
model based on the probabilities p and the scaling factors r. This function
plots five functions tHqL, Dq HqL, aHqL, f HqL, and f HaL .
906
1.2 Glossary
A.2.12 Penning Trap
Motion of two ions in a Penning trap.
Х PenningCMPlot
PenningCMPlot[x0_,y0_,x0d_,y0d_,w_] gives a graphical represen- tation
of the center of mass motion for two ions in the Penning trap. The plot is
created for a fixed cyclotron frequency w in cartesian coordinates Hx, y, zL.
x0 , y0 , x0 d, and y0 d are the initial conditions for integration.
Х PenningI
PenningI[r0_,z0_,e0_,n_,l_,te_] determines the numerical solution of the
equation of motion for the relative components. To integrate the equations
of motion, the initial conditions r0 = rHt = 0L, z0 = zHt = 0L, and the total
energy e0 are needed as input parameters. The momentum with respect to
the r-direction is set to pr0 = 0. Parameters l and n determine the shape of
the potential. The last argument te specifies the endpoint of the integration.
A.2.13 Perihelion Shift
Perihelion shift of a planet.
Х AngularMomentum
AngularMomentum[minorAxes_,majorAxes_,mass_] calculates the angular momentum of a planet.
Х D0Orbit
D0Orbit[planet_String,phiend_,options___] plots the orbit in the case of
vanishing determinants (see text).
Х Energy
Energy[minorAxes_,majorAxes_,mass_] calculates the energy of a planet.
A. Appendix
907
Х orbit
orbit[phiend_,minorAxes_,majorAxes_,mass_]
creates
a
graphical
representation of the perihelion shift if the major and minor axes and the
mass are given.
Х Orbit
Orbit[planet_String] creates a graphical representation of the perihelion
shift for the planets contained in the database.
Х PerihelionShift
PerihelionShift[minorAxes_,majorAxes_,mass_] calculates the numeri- cal
value of the perihelion shift.
Х Planets
Planets[planet_String] creates a list of data for planets and planetoids
stored in the database of the package PerihelionShift. The database
contains the names of the planets, their major axes, their eccentricities, and
the mass of the central planet. Planets['List'] creates a list of the planets in
the data base. Planets['name'] delivers the data of the planet given in the
argument.
A.2.14 Point Charges
Fields of point charges.
Х EnergyDensity
EnergyDensity[coordinates_List] calculates the density of the energy for
an ensemble of point charges. The cartesian coordinates are lists in the
form of {{x,y,z,charge},{...},...}.
Х Field
908
1.2 Glossary
Field[coordinates_List] calculates the electric field for an ensemble of
point charges. The cartesian coordinates are lists in the form
{{x,y,z,charge},{...},...}.
Х FieldPlot
FieldPlot[coordinates_List,type_,options___] creates a contour plot for an
ensemble of point charges. The plot type (Potential, Field, or Density) is
specified as a string in the second input variable. The third argument
allows a change of the Options of ContourPlot and PlotGradientField.
Х Potential
Potential[coordinates_List] creates the potential of an assembly of point
charges. The cartesian coordinates of the locations of the charges are given
in the form of {{x,y,z,charge},{x,y,z,charge},...}.
A.2.15 Poisson Bracket
Canonical Poisson bracket.
Х PoissonBracket
PoissonBracket[a_, b_, q_List, p_List] calculates the Poisson bracket for
two functions a and b which depend on the variables p and q. Example:
PoissonBracket[q,p,{q},{p}] calculates the fundamental bracket relation
between the coordinate and momentum.
A.2.16 Quantum Well
Quantum well in one dimension.
Х PsiASym
PsiASym[x_,k_,a_] determines the antisymmetric eigenfunction for a
potential well of depth -V0 . The input parameter k fixes the energy and 2 a
A. Appendix
909
the width of the well. PsiASym is useful for a numerical representation of
eigenfunctions.
Х PsiSym
PsiSym[x_,k_,a_] determines the symmetric eigenfunction for a potential
well of depth -V0 . The input parameter k fixes the energy and 2 a the
width of the well. PsiSym is useful for a representation of eigenfunctions.
Х Spectrum
Spectrum[V0_,a_] calculates the negative eigenvalues in a potential well.
V0 is the potential depth and 2 a the width of the well. The eigenvalues are
returned as a list and are available in the variables lsym and lasym as
replacement rules. The corresponding eigenfunctions are stored in the
variables Plsym and Plasym. The determining equation for the
eigenvalues is plotted.
A.2.17 Renormalization
Renormalization and percolation.
Х Dim
Dim[n_] calculates the fractal dimension for the critical probability pc .
The dimension depends on m where 1 ╖ m ╖ n - 2, n is the size of the
block used.
Х Nc
Nc[n_] determines the mean number of atoms at the probability pc if m is
changed in the range 1 ╖ m ╖ n - 2. The size of the block is determined by
n.
Х Pcrit
910
1.2 Glossary
Pcrit[n_] determines the critical probability for an n Д n grid under the
variation of m where m is the number of empty locations in the grid. The
range of m is 1 ╖ m ╖ n - 2.
Х RenormPlot
RenormPlot[n_,type_String] plots the functions Nc, Dim or Pcrit.
A.2.18 Tree as a Fractal
Fractal tree.
Х Tree
Tree[options___] creates a fractal tree. The options of the function Tree
determine the form of the fractal created. Options are Generation ф 10,
BranchRotation ф 0.65, BranchSkaling ф 0.75, Branch- Thickness ф
0.7, OriginalThickness ф 0.07, BranchColor ф {RGBColor[0,0,0]}.
Example: Tree[BranchColor ф l1, BranchRotation ф 0.3], l1 is a list
created in the package Tree.
A.3 Mathematica Functions
This appendix contains a short description of the Mathematica functions
used in the book. It is a small selection of the approximately 1200
functions available in the Mathematica kernel. The description given does
not replace the text of the handbook by S. Wolfram ([1.1]).
The first few items describe the use of the shorthand notation of symbols
frequently used in the programming examples. The Mathematica functions
used in the programs and in the notebooks follow.
Х lhs = rhs evaluates rhs and assigns the result to lhs. From then on, lhs is
replaced by rhs whenever it appears. {l1, l2, ...}= {r1, r2, ...} evaluates
the ri and assigns the results to the corresponding li.
A. Appendix
911
Х lhs ф rhs represents a rule that transforms lhs to rhs.
Х expr /. rules applies a rule or list of rules to transform each subpart of an
expression expr.
Х lhs := rhs assigns rhs to be the delayed value of lhs. rhs is maintained in
an unevaluated form. When lhs appears, it is replaced by rhs, evaluated
afresh each time.
Х lhs :> rhs represents a rule that transforms lhs to rhs, evaluating rhs only
when the rule is used.
Х lhs == rhs returns True if lhs and rhs are identical.
Х expr //. rules repeatedly performs replacements
changes.
until expr no longer
Х AppendTo[s, elem] appends elem to the value of s and resets s to the
result.
Х Apply[f, expr] or f @@ expr replaces the head of expr by f. Apply[f,
expr, levelspec] replaces heads in parts of expr specified by levelspec.
Х ArcSin[z] gives the arc sine of the complex number z.
Х ArcTan[z] gives the inverse tangent of z. ArcTan[x, y] gives the inverse
tangent of y/x, where x and y are real, taking into account the quadrant in
which the point (x, y) is located.
Х Begin[ "context`"] resets the current context.
Х BeginPackage[ "context`"] makes context` and System` the only active
contexts. BeginPackage[ "context` ",{"need1` "}, { "need2` "},...}] calls
Needs on the needi.
Х BesselJ[n, z] gives the Bessel function of the first kind J(n, z).
Х Block[{x, y, ...}, expr] specifies that expr is to be evaluated with local
values for the symbols x, y, ... . Block[{x = x0, ...}, expr] defines initial
912
1.3 Mathematica Functions
local values for x,... . Block[{vars}, body /; cond] allows local variables to
be shared between conditions and function bodies.
Х C[i] is the default form for the ith constant of integration produced in
solving a differential equation with DSolve.
Х Chop[expr] replaces approximate real numbers in expr that are close to
zero by the exact integer 0. Chop[expr, tol] replaces with 0 approximate
real numbers in expr that differ from zero by less than tol.
Х Circle[{x, y}, r] is a two-dimensional graphics primitive that represents a
circle of radius r centered at the point {x, y}. Circle[{x, y}, {rx, ry}] yields
an ellipse with semiaxes rx and ry. Circle[{x, y}, r, {theta1, theta2}]
represents a circular arc.
Х Clear[symbol1, symbol2, ... ] clears values and definitions of the
specified symbols. Clear["pattern1", "pattern2", ...] clears values and
definitions of all symbols whose names match any of the specified string
patterns.
Х Coefficient[expr, form] gives the coefficient of form in the polynomial
expr. Coefficient[expr, form, n] gives the coefficient of formn in expr.
Х ContourPlot[f, {x, xmin, xmax}, {y, ymin, ymax}] generates a contour
plot of f as a function of x and y.
Х Cos[z] gives the cosine of z.
Х Cosh[z] gives the hyperbolic cosine of z.
Х Cot[z] gives the cotangent of z.
Х D[f, x] gives the partial derivative of f with respect to x. D[f, {x, n}] gives
the nth partial derivative with respect to x. D[f, x1, x2, ...] gives a mixed
derivative.
Х f' represents the derivative of a function f of one argument. Derivative[n1,
n2, ...][f] is the general form, representing a function obtained from f by
A. Appendix
913
differentiating n1 times with respect to the first argument, n2 times with
respect to the second argument, and so on.
Х Det[m] gives the determinant of the square matrix m.
Х Disk[{x, y}, r] is a two-dimensional graphics primitive that represents a
filled disk of radius r centered at the point {x, y}. Disk[{x, y}, 8rx , r y }]
yields an elliptical disk with semiaxes rx and rx . Disk[{x, y}, r, 8q1 , q2 }]
represents a segment of a disk.
Х Display[channel, graphics] writes graphics or sound to the specified
output channel.
Х Do[expr, {imax}] evaluates expr imax times. Do[expr, {i, imax}]
evaluates expr with the variable i successively taking on the values 1
through imax (in steps of 1). Do[expr, {i, imin, imax}] starts with i = imin.
Do[expr, {i, imin, imax, di}] uses steps di. Do[expr,{i, imin, imax}, {j,
jmin, jmax},... ] evaluates expr looping over different values of j, etc. for
each i. Do[] returns Null, or the argument of the first Return it evaluates.
Х DSolve[eqn, y[x], x] solves a differential equation for the functions y[x],
with independent variable x. DSolve[{eqn1, eqn2, ...},{y1[x1], ...}, {x1,
...}] solves a list of differential equations.
Х Dt[f, x] gives the total derivative of f with respect to x. Dt[f] gives the total
differential of f. Dt[f, {x, n}] gives the nth total derivative with respect to
x. Dt[f, x1, x2, ...] gives a mixed total derivative.
Х EllipticK[m] gives the complete elliptic integral of the first kind K(m).
Х End[ ] returns the present context, and reverts to the previous one.
Х EndPackage[ ] restores $Context and $ContextPath to their values before
the preceding BeginPackage, and prefixes the current context to the list
$ContextPath.
Х lhs == rhs returns True if lhs and rhs are identical.
914
1.3 Mathematica Functions
Х Evaluate[expr] causes expr to be evaluated, even if it appears as the
argument of a function whose attributes specify that it should be held
unevaluated.
Х Exp[z] is the exponential function.
Х Expand[expr] expands products and positive integer powers in expr.
Expand[expr, patt] avoids expanding elements of expr which do not
contain terms matching the pattern patt.
Х FindRoot[lhs == rhs, {x, x0 }] searches for a numerical solution to the
equation lhs == rhs, starting with x = x0 .
Х Flatten[list] flattens out nested lists. Flatten[list, n] flattens to level n.
Flatten[list, n, h] flattens subexpressions with head h.
Х Floor[x] gives the greatest integer less than or equal to x.
Х FontForm[expr, {"font", size}] specifies that expr should be printed in
the specified font and size.
Х Function[body] or body& is a pure function. The formal parameters are #
(or #1), #2, etc. Function[x, body] is a pure function with a single formal
parameter x. Function[{x1, x2,...}, body] is a pure function with a list of
formal parameters. Function[{x1, x2, ...}, body, {attributes}] has the given
attributes during evaluation.
Х <<name reads in a file, evaluating each expression in it, and returning the
last one. Get["name ", key] gets a file that has been encoded with a certain
key.
Х Graphics[primitives, options] represents a
image.
two-dimensional graphical
Х GraphicsArray[{g1, g2, ...}] represents a row of graphics objects.
GraphicsArray[{{g11, g12, ...}, ...}] represents a two-dimensional array of
graphics objects.
A. Appendix
915
Х HermiteH[n, x] gives the nth Hermite polynomial.
Х Hold[expr] maintains expr in an unevaluated form.
Х Hue[h] specifies that graphical objects which follow are to be displayed, if
possible, in a color corresponding to hue h. Hue[h, s, b] specifies colors in
terms of hue, saturation, and brightness.
Х If[condition, t, f] gives t if condition evaluates to True, and f if it
evaluates to False. If[condition, t, f, u] gives u if condition evaluates to
neither True nor False.
Х Im[z] gives the imaginary part of the complex number z.
Х Infinity is a symbol that represents a positive infinite quantity.
Х Input[ ] interactively reads in one Mathematica expression.
Input["prompt"] requests input, using the specified string as a prompt.
Х Integrate[f,x] gives the indefinite integral of f with respect to x.
Integrate[f,{x,
xmin,xmax}]
gives
the
definite
integral.
Integrate[f,{x,xmin,xmax},{y,ymin,ymax}] gives a multiple integral.
Х InterpolatingFunction[range, table] represents an approximate function
whose values are found by interpolation.
Х JacobiAmplitude[u, m] gives the amplitude for Jacobi elliptic functions.
Х JacobiSN[u, m] gives the Jacobi elliptic function sn at u for the parameter
m.
Х Join[list1, list2,... ] concatenates lists together. Join can be used on any set
of expressions that have the same head.
Х LaguerreL[n, x] gives the nth Laguerre polynomial. LaguerreL[n, a, x]
gives the nth generalized Laguerre polynomial.
Х LegendreP[n, x] gives the nth Legendre polynomial. LegendreP[n, m, x]
gives the associated Legendre polynomial.
916
1.3 Mathematica Functions
Х Length[expr] gives the number of elements in expr.
Х Limit[expr, x ф x0 ] finds the limiting value of expr when x approaches x0 .
Х Line[{pt1, pt2,...}] is a graphics primitive which represents a line joining a
sequence of points.
Х {e1, e2, ...} is a list of elements.
Х ListPlot[{y1, y2, ...}] plots a list of values. The x coordinates for each
point are taken to be 1, 2, ... . ListPlot[{{x1, y1}, {x2, y2}, ...}] plots a list
of values with specified x and y coordinates.
Х Log[z] gives the natural logarithm of z (logarithm to base E). Log[b, z]
gives the logarithm to base b.
Х Map[f, expr] or f /@ expr applies f to each element on the first level in
expr. Map[f, expr, levelspec] applies f to parts of expr specified by
levelspec.
Х MapAt[f, expr, n] applies f to the element at position n in expr. If n is
negative, the position is counted from the end. MapAt[f, expr, {i, j, ...}]
applies f to the part of expr at position {i, j, ...}. MapAt[f, expr, {{i1,
j1,...}, {i2, j2, ...}, ...}] applies f to parts of expr at several positions.
Х MatrixForm[list] prints the elements of list arranged in a regular array.
Х Max[x1, x2, ...] yields the numerically largest of the xi. Max[{x1, x2, ...},
{y1, ...}, ... ] yields the largest element of any of the lists.
Х Min[x1, x2, ...] yields the numerically smallest of the xi. Min[{x1, x2,
...}, {y1,...},...] yields the smallest element of any of the lists.
Х Mod[m, n] gives the remainder on division of m by n. The result has the
same sign as n.
Х N[expr] gives the numerical value of expr. N[expr, n] does computations
to n-digit precision.
A. Appendix
917
Х NDSolve[eqns, y, {x, xmin, xmax}] finds a numerical solution to the
differential equations eqns for the function y with the independent variable
x in the range xmin to xmax. NDSolve[eqns, {y1, y2,...}, {x, xmin,
xmax}] finds numerical solutions for the functions yi. NDSolve[eqns, y,
{x, x1, x2, ...}] forces a function evaluation at each of x1, x2, ... . The
range of numerical integration is from Min[x1, x2, ...] to Max[x1, x2,...].
Х Needs["context` ", "file"] loads file if the specified context is not already
in $Packages. Needs["context`"] loads the file specified by
ContextToFilename["context`"] if the specified context is not already in
$Packages.
Х Nest[f, expr, n] gives an expression with f applied n times to expr.
Х NestList[f, expr, n] lists the results of applying f to expr 0 through n times.
Х NIntegrate[f, {x, xmin, xmax}] gives a numerical approximation to the
integral of f with respect to x over the interval xmin to xmax.
Х Normal[expr] converts expr to a normal expression, from a variety of
special forms.
Х NSolve[eqns, vars] attempts to solve numerically an equation or set of
equations for the variables vars. Any variable in eqns but not vars is
regarded as a parameter. NSolve[eqns] treats all variables encountered as
vars above. NSolve[eqns, vars, prec] attempts to solve numerically the
equations for vars using prec digits precision.
Х Off[symbol::tag] switches off a message, so that it is no longer printed.
Off[s] switches off tracing messages associated with the symbols. Off[m1,
m2, ...] switches off several messages. Off[ ] switches off all tracing
messages.
Х On[symbol::tag] switches on a message, so that it can be printed. On[s]
switches on tracing for the symbol s. On[m1, m2, ...] switches on several
messages ma, m2, ... . On[ ] switches on tracing for all symbols.
Х ParametricPlot[{fx, fy}, {t, tmin, tmax}] produces a parametric plot with
x and y coordinates fx and fy generated as a function of t.
918
1.3 Mathematica Functions
ParametricPlot[{{fx, fy}, {gx, gy}, ...}, {t, tmin, tmax}] plots several
parametric curves.
Х ParametricPlot3D[{fx, fy, fz}, {t, tmin, tmax}]
produces a
three-dimensional space curve parameterized by a variable t which runs
from tmin to tmax. ParametricPlot3D[{fx, fy, fz}, {t, tmin, tmax}, {u,
umin, umax}] produces a three-dimensional surface parameterized by t and
u. ParametricPlot3D[{fx, fy, fz, s}, ...] shades the plot according to the
color specifications. ParametricPlot3D[{{fx, fy, fz}, {gx, gy, gz}, ...}, ...]
plots several objects together.
Х expr[[i]] or Part[expr, i] gives the ith part of expr. expr[[-i]] counts from
the end. expr[[0]] gives the head of expr. expr[[i, j, ...]] or Part[expr, i, j,
...] is equivalent to expr[[i]][[j]] ... . expr[[ {i1, i2, ...}]] gives a list of the
parts i1, i2, ... of expr.
Х Partition[list, n] partitions list into non-overlapping sublists of length n.
Partition[list, n, d] generates sublists with offset d. Partition[list, {n1, n2,
...}, {d1, d2, ...}] partitions successive levels in list into length ni sublists
with offsets di.
Х Pi is pi, with numerical value 3.14159... .
Х Plot[f, {x, xmin, xmax}] generates a plot of f as a function of x from xmin
to xmax. Plot[{f1, f2, ...}, {x, xmin, xmax}] plots several functions fi.
Х x + y + z represents a sum of terms.
Х Point[coords] is a graphics primitive that represents a point.
Х x y gives x to the power y.
Х PowerExpand[expr] expands nested powers, powers of products,
logarithms of powers, and logarithms of products. PowerExpand[expr,{x1,
x2,...}] expands expr with respect to the x1. Use PowerExpand with
caution because PowerExpand does not pay attention to branch cuts.
Х Print[expr1, expr2,... ] prints the expri, followed by a newline (line feed).
A. Appendix
919
Х Protect[s1, s2, ... ] sets the attribute Protected for the symbols si. Protect[
"form1", "form2 ", ...] protects all symbols whose names match any of the
string patterns formi.
Х Quit[ ] terminates a Mathematica session.
Х Random[ ] gives a uniformly distributed pseudorandom Real in the range
0 to 1. Random[type, range] gives a pseudorandom number of the
specified type, lying in the specified range. Possible types are Integer,
Real, and Complex. The default range is 0 to 1. You can give the range
{min, max} explicitly; a range specification of max is equivalent to {0,
max}.
Х Re[z] gives the real part of the complex number z.
Х ReleaseHold[expr] removes Hold and HoldForm in expr.
Х Replace[expr, rules] applies a rule or list of rules in an attempt to
transform the entire expression expr.
Х expr /. rules applies a rule or list of rules in an attempt to transform each
subpart of an expression expr.
Х expr //. rules repeatedly performs replacements until expr no longer
changes.
Х RGBColor[red, green, blue] specifies that graphical objects which follow
are to be displayed, if possible, in the color given.
Х lhs фrhs represents a rule that transforms lhs to rhs.
Х Save["filename", symb1, symb2, ...] appends the definitions of the
symbols symbi to a file.
Х Series[f, {x, x0 , n}] generates a power series expansion for f about the
point x = x0 to order Hx - x0 Ln . Series[f, {x, x0 , nx}, {y, y0 , ny}]
successively finds series expansions with respect to y, then x.
920
1.3 Mathematica Functions
Х Show[graphics, options] displays two- and three-dimensional graphics,
using the options specified. Show[g1, g2, ...] shows several plots
combined. Show can also be used to play Sound objects.
Х Simplify[expr] performs a sequence of transformations on expr and
returns the simplest form it finds.
Х Sin[z] gives the sine of z.
Х Sinh[z] gives the hyperbolic sine of z.
Х Solve[eqns, vars] attempts to solve an equation or set of equations for the
variables vars. Any variable in eqns but not vars is regarded as a
parameter. Solve[eqns] treats all variables encountered as vars above.
Solve[eqns, vars, elims] attempts to solve the equations for vars,
eliminating the variables elims.
Х Sort[list] sorts the elements of list into canonical order. Sort[list, p] sorts
using the ordering function p.
Х SphericalHarmonicY[l, m, theta, phi] gives the spherical harmonic
Yl,m (q, f).
Х Sqrt[z] gives the square root of z.
Х Sum[f, {i, imax}] evaluates the sum of f with i running from 1 to imax.
Sum[f, {i, imin, imax}] starts with i = imin. Sum[f, {i, imin, imax, di}]
uses steps di. Sum[f, {i, imin, imax}, {j, jmin, jmax},...] evaluates a
multiple sum.
Х Table[expr, {imax}] generates a list of imax copies of expr. Table[expr,
{i, imax}] generates a list of the values of expr when i runs from 1 to imax.
Table[expr, {i, imin, imax}] starts with i = imin. Table[expr, {i, imin,
imax, di}] uses steps di. Table[expr, {i, imin,
imax}, {j, jmin,
jmax},...] gives a nested list. The list associated with i is outermost.
Х Take[list, n] gives the first n elements of list. Take[list, -n] gives the last n
elements of list. Take[list, {m, n}] gives elements m through n of list.
A. Appendix
921
Х Tan[z] gives the tangent of z.
Х Text[expr, coords] is a graphics primitive that represents text
corresponding to the printed form of expr, centered at the point specified
by coords.
Х Thread[f[args]] ``threads'' f over any lists that appear in args.
Thread[f[args], h] threads f over any objects with head h that appear in
args. Thread[f[args], h, n] threads f over objects with head h that appear in
the first n args. Thread[f[args], h, -n] threads over the last n args.
Thread[f[args], h, {m, n}] threads over arguments m through n.
Х Unprotect[s1, s2, ...] removes the attribute Protected for the symbols si.
Unprotect["form1","form2", ...] unprotects all symbols whose names
textually match any of the formi.
Х Which[test1, value1, test2, value2, ... ] evaluates each of the testi in turn,
returning the value of the valuei corresponding to the first one that yields
True.
References
Volume I
[1]
Chapter 1
[1.1]
S. Wolfram, The Mathematica book, 5th ed.
Media/Cambridge University Press, Cambridge 2003.
[1.2]
M. Abramowitz & I.A. Stegun, Handbook of Mathematical
Functions. Dover Publications, Inc., New York, 1968.
[1.3]
N. Blachman, Mathematica: A Practical Approach. Prentice Hall,
Englewood Cliffs, 1992.
[1.4]
Ph. Boyland, A. Chandra, J. Keiper, E. Martin, J. Novak, M.
Petkovsek, S. Skiena, I. Vardi, A. Wenzlow, T. Wickham-Jones,
D. Withoff, and others, Technical Report: Guide to Standard
Mathematica Packages, Wolfram Research, Inc. 1993.
[2]
Chapter 2
Wolfram
924
References
[2.1]
R. Maeder, Programming in Mathematica. Addison-Wesley Publ.
Comp. Inc., Redwood City, 1991.
[2.2]
L.D. Landau & E.M. Lifshitz, Mechanics. Addison-Wesley,
Reading, Massachusetts, 1960.
[2.3]
J. B. Marion, Classical Dynamics of Particles and Systems.
Academic Press, New York, 1970.
[2.4]
R. Courant & D. Hilbert, Methods of Mathematical Physics, Vol.
1+2. Wiley (Interscience), New York, 1953.
[2.5]
R.H. Dicke, Science 124, 621, (1959)
[2.6]
R.V. EЖtvЖs,Ann.Phys. 59, 354, (1896)
[2.7]
L. Southerns,Proc.Roy.Soc.(London),A, 84, 325, (1910)
[2.8]
P. Zeeman,Proc.Amst.,20,542,(1917)
[2.9]
G. Baumann, Symmetry Analysis of Differential equations using
Mathematica, Springer, New York, (2000).
[2.10]
H. Geiger and E. Marsden, The Laws of Deflexion of a Particles
through Large Angles, Phil. Mag. 25, 605, 1913.
[2.11]
Ph. Blanchard and E. BrЭning, Variational Methods in
Mathematical Physics, Springer, Wien, 1982.
[3]
Chapter 3
[3.1]
F. Calogero & A. Degasperis, Spectral Transform and Solitons:
Tools to solve and investigate nonlinear evolution equations.
North-Holland Publ. Comp., Amsterdam, 1982.
[3.2]
V.A. Marchenko, On the Reconstruction of the Potential Energy
from Phases of the Scattered Waves. Doklady Akademii Nauk
SSSR, 104, 695, 1955.
References
925
R.M. Miura, C. Gardner & M.D. Kruskal. Korteweg-de Vries
equation and generalizations. II Existence of Conservation Laws
and Constants of Motion. Journal of Mathematical Physics 9,
1204, 1968.
[3.4]
T.R. Taha & M.J. Ablowitz, Analytical and numerical solutions of
certain nonlinear evolution equations. I. Analytical. Journal of
Computational Physics 55, 192, 1984.
[3.5]
N.J. Zabusky & M.D. Kruskal, Interactions of 'solitons' in a
collisionless plasma and the recurrence of initial states. Physical
Review Letters 15, 240, 1965.
Volume II
[4]
Chapter 4
[4.1]
G. Arfken, Mathematical Methods for Physicists. Academic Press,
New York, 1966.
[4.2]
P.M. Morse & H. Feshbach, Methods of Theoretical Physics.
McGraw-Hill, New York, 1953.
[4.3]
W. Paul, O. Osberghaus & E. Fischer, Ein IonenkДfig.
Forschungsbericht des Wissenschafts- und Verkehrsministeriums
Nordrhein-Westfalen, 415, 1, 1958.
Similar work has been done by H. G. Dehmelt, Radiofrequency
Spectroscopy of Stored Ions I: Storage, Advances in Atomic and
Molecular Physics 3(1967) 53-72; D. J. Wineland, W.M. Itano and
R.S. van Dyck Jr., High-Resolution Spectroscopy of Stored Ions,
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Index
A
Abel, 941
absolute temprature, 766
ac-field, 610
action, 779
algebraic equation, 986
algorithm, 987, 993
amorphous semiconductor, 997
amplitude, 731
analytical calculation, 545
analytical methods, 906
angle of inclination, 793
angular momentum, 616, 751?752,
786
angular quantum number, 757
anharmonic, 740
anharmonic oscillator, 740
anhilation operator, 738
annihilation operator, 737
anomalous diffusion, 984, 1006
anomalous diffusion exponent, 1006
ansatz, 755
aphelion, 783
apogee, 789
associated Legendre polynomials, 741
assumption, 949
astrophysics, 807
asymptotic circles, 789
asymptotic direction, 794
asymptotic expansion, 747
asymptotic representation, 748
atomic systems, 706
average energies, 803
Avogadro number, 767
Avogadro's constant, 766
axial frequency, 613
B
balls, 903
Barns integral, 983
base angle, 920
Bernoulli, 939
Bessel function, 956
932
Bianchi identities, 803, 811
binding of atoms, 758
black hole, 706
blackbody radiation, 703
blocks, 931
Boltzmann constant, 766?767
borderline, 903
Born, 705
bound region, 803
bound state, 768, 803
boundary, 900
boundary condition, 590
Dirichlet, 600
Dirichlet and von Neumann,
600
von Neumann, 600
boundary line, 905
boundary problem, 598?599
bounded sets, 900
bounded subset, 908
box counting, 906, 908
box counting dimension, 908
box counting method, 905
box dimension, 908, 912
box length, 914
Boyle temperature, 803
Boyle temperaure, 805
Broglie, 704
bronchial tree, 905
C
calculus, 948
Index
Cantor, 906
capacity dimension, 908
Cartesian coordinates, 592
Cartesian metric, 797
Cartesian space, 804
Cauchy's integral formula, 942
center of mass coordinates, 611
center of mass motion, 612
central field, 752
central force, 777
central force field, 751
chain rule, 945, 947
changing scales, 930
chaotic, 617
characteristic function, 924
characteristic polynomial, 613, 783,
792
charge density, 590
charge distribution, 590
charge-free, 600
charged mass point, 822
Christoffel symbols, 801, 805
circular force, 588
classical mechanics, 546, 715
classical orbit, 789
classical probability, 733
classically forbidden, 715
commuting operators, 752
complete basis, 713
complete elliptic integrals, 787
complex field, 707
Index
complex materials, 997
composition rule, 945?946
conducting wall, 609
cones, 903
confluent hypergeometric function,
756
congruence, 919
congruent triangle, 918
continuity condition, 716
continuum state, 768, 803
continuum theory, 599
contour length, 908
contour plot, 592
convolution, 961, 963
convolution type integral, 974
coordinate transformation, 804
correlation length, 935
Coulomb, 588
Coulomb force, 611
Coulomb interaction, 611, 754
count, 912
countable sets, 900
covariant divergence, 823
creation operator, 737
critical exponent, 935?936
critical phenomena, 930
critical point, 930, 935
curvature scalar, 802
curved space, 774?775
cyclotron frequency, 613, 616
cylinders, 903, 908
933
cylindrical coordinates, 806
cylindrical coordinates , 798
D
Davy, 588
dc-potential, 612
Debye process, 995
Debye relaxation, 995
decades, 997
degenerate electronic states, 808
density, 734
derivatives, 963
determinant, 717
diagonal elements, 810
diatomic molecule, 740, 808
diatomic molecules, 807
dielectric relaxation, 997
differential equation, 985?986
differential equations, 964
differentiation of a constant, 949
diffusion constant, 707, 1007
diffusion equation, 707
dimer parition function, 808
Dingle's metric, 812
dipole, 592
Dirac's delta function, 590
Dirichlet boundary condition, 600
Dirichlet problem, 600
discrete spectrum, 602, 745
disjunct boxes, 908
disociation limit, 809
934
dispersion, 708, 712
dispersion force, 767
dispersion relation, 712
dispersive phenomena, 709
dispersive wave, 708
distribution, 972
domain boundaries, 716
driven rubber equation, 1004
dynamic trap, 609
dynamo, 588
E
eccentricity, 786
Eddington-Finkelstein, 809
Eddington-Finkelstein line element,
809
edge length, 909
eigenfunction, 601, 713, 731?732,
739, 743
antisymmetric, 718
symmetric, 718
eigenfunction expansion, 601
eigenstate, 713
eigenvalue, 601, 713, 715
eigenvalue equation, 720
eigenvalue problem, 601, 731, 752
eikonal equation, 707
Einstein tensor, 819
Einstein's field equation, 773
Einstein's field equations, 795, 799,
803
electric field, 590?591
electric force, 588
electric potential, 600
Index
electricity, 588
electromagnetic field, 589
electromagnetic force, 611
electromagnetic phenomena, 590
electronic degeneracy, 808
electrostatic, 590
electrostatic phenomena, 599
ellipse, 777
ellipsoids, 908
elliptic function, 780
energy, 714, 786
energy density, 777
enthalpy, 768, 778
entropy, 768, 778
entropy dimension, 908
equation of state, 769
equilibrium point, 730
Euclidean space, 797
Euler, 941
Euler-Lagrange equations, 779
excitation energy, 808
expansion coefficient, 601
expectation value, 934
exponential, 987
exponential decay, 996
external force, 989
external potential, 707
F
Farady, 588
field, 588
Index
field equations, 801
first formula by Green, 599
first kind Fredholm integral equation,
976
first quantum correction, 780
fit, 916
fixed point, 932
flat space, 805
FlЭgge, 740
focus, 777
Fourier, 941
Fourier transform, 708, 958, 1008
Fox H-function, 968
Fox function, 967, 982?983
fractal, 906, 930
fractal cluster dimension, 935
fractal dimension, 906
fractal geometry, 937
fractals, 546
Fractals, 899
fractional calculus, 937
fractional derivative, 943
fractional derivatives, 940, 943
fractional differential equations, 984
fractional differentiation, 937, 943,
949
fractional dimension, 900
fractional integral, 953
fractional integral equation, 959
fractional relaxation equation, 995
FractionalCalculus, 949
Fredholm convolution integral, 972
935
Fredholm equation, 973
Fredholm integral equation, 979, 998
free particle, 709
Friedman, 774
fundamental force, 706
G
G-function, 939, 964
gas, 930
gas constant, 766
gas imperfection, 769
gauge conditions, 804
Gauъ, 938
Gaussian behavior, 1006
Gaussian coordinates, 804
Gauss's law, 590
Gauss's theorem, 599
general relativity, 773
generalized diffusion equation, 1007
generalized dimension, 924, 926
generalized hypergeometric function,
967
generalized Mittag-Leffler function,
998
generalized relaxation equation, 991
generating operator, 737
geometric complexity, 900
geometric mass, 827
geometric structure, 899
geometrical objects, 903
Gibb's techniques, 766
gravitation, 599
gravitation phenomena, 775
gravitational collapses, 774
936
gravitational constant, 778
gravitational field, 777
gravitational radiation, 774
Green's, first formula, 600
second formula, 600
Green's function, 590, 599, 605, 708
ground electronic state, 809
ground state, 737
H
H-atom, 751
Hamiltonian, 730, 751
Hamiltonian operator, 714
Hankel transform, 959
harmonic external force, 1004
harmonic function, 613
harmonic oscillations, 730
harmonic oscillator, 613, 712, 729
Hausdorff, 900
heat capacity, 778
Heisenberg, 705
Hermite, 732
Hermite polynomial, 732, 737
high frequency limit, 703
high temperature chemistry, 807
HЖlder exponent, 925?926
hydrodynamics, 599
hydrogen atom, 755
hyper-geometric function, 745
hypergeometric function, 732, 772,
952
hypergeometric functions, 793
Index
I
induction, 588
information dimension, 908
inhomogeneous field equations, 822
initial condition, 708, 1007
initial value problem, 986?987
integral equation, 973, 975, 990
integral equations, 964, 972
integral theorem of Gauss, 600
integral transform, 958, 991
integral transforms, 986
intermolecular force, 771
intermolecular potential, 766
internal erenrgy, 774
internuclear distance, 769
invariant, 930
inverse metric tensor, 808
inverse scattering method, 740
inverse temperature, 772
InverseMellinTransform[], 966
ion trap, 609
isotropic, 800
J
Jones, 767
Jordan, 705
Joul-Thomson coefficient, 778
K
Kannerligh Onnes, 765
Kepler, 777, 789
kernel, 959, 975
Index
Kerr solution, 827
Kihara potential, 769?770
Koch, 906
Koch curve, 918?919
Koch snowflake, 906
Kohlrausch-William-Watts, 971
Kolmogorov entropy, 908
Kruskal coordinates, 818
Kruskal solution, 818
Kruskal variables, 822
Kummer's differential equation, 756
Kummer's function, 757
L
Lacroix, 941
Lagrangian, 617, 778
Laguerre polynomial, 757
Laguerre's function, 757
Langevin equation, 985
Laplace equation, 598, 609
cylindrical coordinates, 603
Laplace integral equation, 978
Laplace space, 987
Laplace transform, 771, 959,
986?987, 991
large molecule, 740
lattice, 931
Lebesgue, 900
Lebesgue measure, 900
Legendre function, 743, 753
Legendre polynomial, 741
Legendre transform, 925
937
Leibniz, 938
Leibniz rule, 945
Leibniz's rule, 947
length, 920
length of a border, 899
Lennard, 767
Lennard-Jones potential, 767, 769
Lenz vector, 777
L`Hospital, 938
light bending, 790
light ray, 790
light rays, 791
line element, 795, 804, 920
linear displacement, 740
linear first-order ODE, 985
linear fractional differential equation,
990
linearity, 708, 945, 990
Liouville, 939, 942
Liouville fractional integral, 943
liquid, 930
local minimum, 729
log-log plot, 906, 909
London, 767
Lorentz force, 611
Lotmar, 740
low frequency limit, 703
M
macroscopic thermodynamics, 765
magnetic field, 610
magnetic force, 588
938
magnetic quantum number, 753
magnetism, 588
major semi axis, 786
Mandelbrot, 899, 925
Mandelbrot set, 901
mapping, 901
mass density, 777
mathematical calculation, 545
matrix algebra, 705
matrix mechanics, 705
Maxwell, 588
Maxwell tensor, 823
Maxwell's equations, 822
mean square displacement, 1006
mean value, 707
measurement, 713
Meijer G-function, 968
Mellin representation, 994
Mellin space, 975, 992
Mellin transform, 958?960, 973, 975,
979, 991
Mellin-Barns integral, 994
MellinTransform[], 961
memory, 998
memory kernel, 1007
memory-diffusion equation, 1007
Mercury, 777, 785
mesh-size, 905, 934
metastable state, 768, 803
metric, 795
metric dimension, 908
Index
metric geodesics, 801
metric tensor, 795, 798?799, 801
microscopic physics, 765
Minkowski space, 799
Mittag-Leffler function, 952, 993
modulus, 794
molecular interactions, 766
molecular orbital, 758
molecular potential, 803
moments, 972
momentum space, 737
monoatomic assembly, 769
monomer partition function, 808
monster curves, 899
movement of perihelion, 775
multi-fractal, 924, 926
multi-fractal characteristic, 926
multi-fractal distribution, 925
multi-Fractals, 923
N
nth-order ODE, 985
nano phenomena, 706
natural objects, 899, 905
negative second-order derivative, 942
Newton, 611, 775, 777, 938
non-commutative algebra, 705
non-degenerate, 733
non-integer derivatives, 938
nonlinear evolution equation, 740
normal gradient, 600
Index
normalization, 716
normalize, 709
normalized solution, 752
null geodesic, 790
O
option, 951
orbit, 780
orbital, 764
orbital motion, 777
Ornstein, 766
orthogonal, 601
P
paraboloid, 609
parameterized curve, 801
partition function, 768, 807
Paul, 609
Peano, 906
Penning, 609
Penning trap, 609
percolation cluster, 931?932
percolation theory, 931
perfect gas, 768
perihelion, 777, 783
perihelion rotation, 777
perihelion shift, 777, 785
period, 730, 783
perturbation theory, 936
phase diagram, 930
phase transition, 932
phase transitions, 930
939
physical characteristics, 900
Planck, 703
Planck constant, 707
plane filling, 906, 921
plane wave, 708
planetary system, 777
point charge, 591
Poisson equation, 590
polymer, 984
polymer science, 931
polynomial, 732
porous medium, 931
PЖschel, 740
PЖschel-Teller potential, 740
potential, 590?591
potential barrier , 734
potential depth, 743
potential well, 714
power law, 937, 997
pressure, 803
pressure equilibrium constant, 808
principal quantum number, 757
probability, 707, 923
probability amplitude, 705
probability distribution, 710, 733
projection plane, 904
properties of the Mellin transform,
960
Pythagoras, 918
Q
quadruple, 595
940
quadrupole field, 609, 611
quantum chemistry, 740
quantum correction, 767, 778
quantum corrections, 767
quantum dot, 751
quantum dot model, 707
quantum mechanical corrections, 778
quantum mechanical operators, 731
quantum mechanical state, 737
quantum mechanics, 546, 704, 707
quantum number, 753, 757, 807
quasi elliptic orbits, 783
Index
relaxation equation, 986, 989
relaxation of polymers, 997
relaxation oscillation equation, 1000
relaxation phenomenon, 984
relaxation time, 986
relaxation time spectrum, 899
renormalization, 930
renormalization error, 936
renormalization group, 929?930
renormalized lattice, 931
repulsive branch, 804
resolution transformation, 929
rest mass, 777
Ricci scalar, 802?803
R
radial quantum number, 757
Ricci scalar , 825
radial wave function, 754
Ricci tensor, 801?803
random force, 985
Riemann, 775, 939, 942
random links, 931
Riemann fractional integral, 943
random number, 909
Riemann geometry, 795
rational function, 964
Riemann tensor, 801?802
Rayleigh, 703
Riemann tensor , 807
reaction kinetics, 807
Riemann z-function, 965
real gas, 766
Riemann-Liouville fractional integral,
reduced de Broglie wavelength, 789
943
reduced mass, 807
Riemann-Liouville operator, 945
reduced quantities, 793
RiemannLiouville[], 948
reflection coefficient, 747
RiemannLiouville[], 944
regularity, 604
Riemann's theory, 774
Reissner-Nordstrom solution, 773, 822 rosette, 784
relative coordinates, 611
rosettes, 777
relative motion of the ions, 615
rotating black hole, 827
Index
rotation-vibration eigenfunction, 807
rotation-vibration SchrЖdinger
equation, 807
rotational barrier, 807
Rydberg-diatomic potential, 768
S
scaling, 616, 731, 961
scaling behavior, 918
scaling exponent, 909, 916
scaling factor, 920, 926
scaling factors, 923
scaling property, 962
scaling range, 909
scaling transformation, 930
scattering problem, 748
SchrЖdinger, 704
SchrЖdinger equation, 707, 740, 752
Schwarzschild, 774
Schwarzschild line element, 810
Schwarzschild metric, 778, 790
Schwarzschild radius, 778, 791
Schwarzschild solution, 773, 799, 809
second formula by Green, 600
second kind of Fredholm equation,
979
second quantum correction, 780
second virial coefficient, 765?766,
769, 793
secular equation, 617
self-similar, 909
self-similarity, 903, 906, 918, 923
semi fractional derivative, 957
semi-group, 930
941
semiclassical expansion, 767
semiconductors, 706
semifractional differential equation,
1002
separation, 604
shifting, 961
shifting property, 962
singular, 810
singularity, 783
slope, 906
slow decay, 1000
small oscillations, 730
snowflake, 900
space time, 795
specific heat, 768
spectral density, 708, 712
spectral properties, 712
spectroscopic dissociation energy,
809
spectrum, 926
spheres, 908
spherical coordinates, 798, 807
spherical Einstein equations, 775
spherical symmetry, 799, 809, 822
spherically symmetric, 751
spring constant, 730
standard diffusion, 1007
standard relaxation, 995
static magnetic field, 611
static trap, 609
stationary SchrЖdinger equation, 745
statistical physics, 599
942
straight line, 903
straight lines, 903
super lattice, 931, 934
superposition, 707?708, 764, 945, 991
symmetric difference, 925
symmetry, 754
syntax, 545
T
Teller, 740
template, 948
thermodynamic function, 767
thermodynamics, 599, 703
thought experiment, 775
total energy, 715
total potential, 600
transcendent equation, 720
transcendental functions, 952
transmission coefficient, 747
tree, 904
tunneling, 734
turning point, 734
two ions, 612
U
uncertainty principle, 705
unification, 706
unstable, 933
V
vacuum case, 799
vacuum equations, 803
vacuum field equations, 800
Index
Van-der-Waals equation, 766
variational principle, 779
velocity of light, 777
vibrational state, 809
viral coefficient, 766
viral equation of state, 766
virial coefficient, 769
virial coefficients, 767
virial equation, 765?766
virial equation , 767
Volterra, 990
von Neumann boundary condition,
600
W
wave, 959
wave function, 707, 712?713, 732,
734, 758
wave mechanics, 704
wave packet, 708?709
Weierstrass, 906
Weierstrass function, 783, 791
well depth, 720, 769
Weyl, 939
Wien, 703
world time, 800
Y
yardstick, 904
yardstick method, 905, 908
Yukawa particle, 751
t["
Print["
Print["
Print["
Print["
Print["
Print["
Koch
:
QKoch :
VKoch :
WKoch :
Quad
:
Star
:
Square :
N-gon :
Mixture:
Random :
Koch curve"];
rectangular Koch curve"];
altered Koch curve"];
variable angle Koch curve"];
rectangular curve"];
Koch star"];
Koch square"];
Koch N gon"];
2 x Koch and 2 x QKoch"];
random generation"]];
(* --- plot the Koch curves --- *)
If[curve == "Koch" ||
curve == "QKoch" ||
curve == "VKoch" ||
curve == "WKoch" ||
curve == "Quad",
(* --- ToExpression transforms a string to an
expression --- *)
k1 =
ToExpression[curve][Line[{{0,0},{1,0}}]];
Do[
k1 = docurve[ToExpression[curve],k1],
{k,1,generations}];
FractalPlot[k1]
];
(* --- plot a Koch star --- *)
If[curve == "Star",
corners = 3;
k1 = NGon[Line[{{0,0},{1,0}}]];
Do[
k1 = docurve[Koch,k1],
{k,1,generations}];
FractalPlot[k1]
];
(* --- plot a Koch square --- *)
If[curve == "Square",
corners = 4;
k1 = NGon[Line[{{0,0},{1,0}}]];
Do[
k1 = docurve[Koch,k1],
{k,1,generations}];
FractalPlot[k1]
];
(* --- plot a Koch N gon --- *)
If[curve == "N-gon",
k1 = NGon[Line[{{0,0},{1,0}}]];
Do[
892
7.8 Packages and Programs
k1 = docurve[Koch,k1],
{k,1,generations}];
FractalPlot[k1]
];
(* --- plot a mixture of Koch curves --- *)
If[curve == "Mixture",
k1 = Koch[Line[{{0,0},{1,0}}]];
k1 = docurve[Koch,k1];
k1 = docurve[QKoch,k1];
k1 = docurve[QKoch,k1];
FractalPlot[k1]
];
(* --- plot a random sequence of Koch curves --- *)
If[curve == "Random",
listec ={Koch,QKoch,VKoch,WKoch,Quad,NGon};
k2 = Random[Integer,{1,6}];
k3 = Random[Integer,{1,6}];
If[k2 == 6 || k3 == 6, corners =
Random[Integer,{3,12}]];
name1 = listec[[k2]];
name2 = listec[[k3]];
k1 = name1[Line[{{0,0},{1,0}}]];
k1 = docurve[name1,k1];
Do[
k1 = docurve[name2,k1],
{k,1,generations-1}];
FractalPlot[k1]
];
];
End[];
EndPackage[];
7.8.3 Multifactals
The multifractal package provides functions for the determination of
multifractal spectra.
BeginPackage@"MultiFractal`"D;
Clear@Dq, Tau, Alpha, MultiFractalD;
MultiFractal::usage =
"MultiFractal@p_List,r_ListD calculates the
multifractal spectrum D_q for a model
7. Fractals
893
based on the probabilities p and the
scaling factors r. This function plots
five functions TauHqL, D_qHqL,
AlphaHqL, fHqL and fHAlphaL.";
Begin@"`Private`"D;
Hcalculate the multifractal dimensionsL
Dq@p_List, r_ListD := Block@8l1, l2, listrg = 8<<,
Hlength of the listsLl1 = Length@pD;
l2 = Length@rD;
If@l1 m l2,
Hvariation of q and determination of D_qL
Do@gl1 = Sum@p@@jDD ^ q r@@jDD ^ HHq 1L DfractalL,
8j, 1, l1<D 1;
result = FindRoot@gl1 m 0, 8Dfractal, 3, 3<D;
result = Dfractal Й. result;
Hcollect the results in a listL
AppendTo@listrg, 8q, result<D,
8q, 10, 10, .101<D, Print@" "D;
Print@" Lengths of lists are different!"D;
listrg = 8<D;
listrgD;
Hcalculate TauL
Tau@result_ListD := Block@8l1, listtau = 8<<,
Hlengths of the listsL
l1 = Length@resultD;
Hcalculate TauLDo@
AppendTo@listtau, 8result@@k, 1DD, result@@k, 2DD
H1 result@@k, 1DDL<D, 8k, 1, l1<D;
listtauD;
HLegendre transformL
Alpha@result_ListD := Block@8l1, dq, listalpha = 8<,
listf = 8<, listleg = 8<, mlist = 8<, pl1, pl2<,
Hlengths of the listsL
l1 = Length@resultD;
Hdetermine the differential dqL
894
7.8 Packages and Programs
dq = Hresult@@2, 1DD result@@1, 1DDL 2;
Hcalculate Alpha by numerical
differentiationLDo@
AppendTo@listalpha, 8result@@k, 1DD,
Hresult@@k + 1, 2DD result@@k 1, 2DDL Й
dq<D, 8k, 2, l1 1<D;
l2 = Length@listalphaD;
Hcalculate f and
collect the results in a listLDo@
AppendTo@listf, 8result@@k, 1DD, Hresult@@k, 1DD
listalpha@@k, 2DD result@@k, 2DDL<D;
listalpha@@k, 2DD = listalpha@@k, 2DD,
8k, 1, l2<D;
Hlist of the Legendre transformsL
Do@AppendTo@listleg,
8listalpha@@k, 2DD, listf@@k, 2DD<D;
AppendTo@mlist, listf@@k, 2DDD, 8k, 1, l2<D;
Hplot f and alpha versus qLpl1 =
ListPlot@listalpha, PlotJoined ▒ True, AxesLabel ▒
8"q", "D"<, Prolog ▒ Thickness@0.001DD;
pl2 = ListPlot@listf, PlotJoined ▒ True, AxesLabel ▒
8"q", "f"<, Prolog ▒ Thickness@0.001DD;
Show@8pl1, pl2<, AxesLabel ▒ 8"q", "D,f"<D;
Hplot the Legendre
transform f versus alphaL
ListPlot@listleg, AxesLabel ▒ 8"D", "f"<D;
Hprint the maximum of f=D_ 0L
maxi = Max@mlistD;
Print@" "D;
Print@"
D_0 = ", maxiDD;
Hcalculate the multifractal propertiesL
MultiFractal@p_List, r_ListD :=
Block@8listDq, listTau<,
Hdetermine D_qLlistDq = Dq@p, rD;
ListPlot@listDq, PlotJoined ▒ True, AxesLabel ▒
8"q", "Dq "<, Prolog ▒ Thickness@0.001DD;
Hcalculate TauLlistTau = Tau@listDqD;
ListPlot@listTau, PlotJoined ▒ True, AxesLabel ▒
8"q", "W"<, Prolog ▒ Thickness@0.001DD;
Hdetermine the Hoelder exponentL
7. Fractals
895
Alpha@listTauDD;
End@D;
EndPackage@D;
7.8.4 Renormalization
This package supports the calculations of renormalization.
BeginPackage["Renormalization`"];
Clear[f, Pcrit, Nc, Dim, RenormPlot];
Nc::usage = "Nc[n_] determines the mean number of
atoms at the probability
p_c if m is changed in the range 1 <= m <= n-2. The
size of the block is
determined by n.";
Dim::usage = "Dim[n_] calculates the fractal
dimension for the critical
probability p_c. The dimension depends on m where 1
<= m <= n-2, n is
the size of the block used.";
Pcrit::usage = "Pcrit[n_] determines the critical
probability for an n x n
grid under the variation of m where m is the number
of empty locations in the
grid. The range of m is 1 <= m <= n-2.";
RenormPlot::usage = "RenormPlot[n_,typ_String] plots
the functions Nc, Dim
or Pcrit.";
Begin["`Private`"];
(* --- auxilary function --- *)
f[p_,n_,m_]:=
Sum[Binomial[n,i]*p^(n-i)*(1-p)^i,{i,0,m}];
(* --- mean number of particles on a grid --- *)
Nc[n_]:= Block[{p, ncliste={}},
p = Pcrit[n];
896
7.8 Packages and Programs
Do[
AppendTo[ncliste,
Sum[Binomial[n,i]*(n-i)*p[[k]]^(n-i-1)*(1-p[[k]])^i,
{i,0,k}]],
{k,1,n-2}];
ncliste
];
(* --- fractal dimension at the critical probability
--- *)
Dim[n_]:=
N[Log[Nc[n]]/Log[Sqrt[n]]];
(* --- critical probability on a n x n grid --- *)
Pcrit[n_]:=Block[{ph, p, erg, erg1, gl1, pliste1={}},
If[n > 2,
Do[
gl1 = p - f[p,n,i];
(* --- solution of the fixpoint equation --- *)
erg = NSolve[gl1==0,p];
erg = p /. erg;
(* --- use only real solutions --- *)
erg1 = {};
Do[If[Head[erg[[k]]]==Real,AppendTo[erg1,erg[[k]]]],
{k,1,Length[erg]}];
(* --- looking for solutions between 0 and 1
--- *)
erg = Sort[erg1];
erg1 = {};
Do[If[erg[[k]] > 0.0 ,
AppendTo[erg1, erg[[k]] ] ],
{k,1,Length[erg]}];
ph = Min[erg1];
AppendTo[pliste1,ph],
{i,1,n-2}],
Print["
"];
Print[" choose n > 2 "]];
pliste1];
(* --- plot the results --- *)
RenormPlot[n_,typ_String]:=Block[{},
If[typ == "Pcrit",
ListPlot[Pcrit[n],AxesLabel->{"m","pc"}],
If[typ == "Nc",
7. Fractals
897
ListPlot[Nc[n],AxesLabel->{"m","Nc"}],
If[typ == "Dim",
ListPlot[Dim[n],AxesLabel->{"m","D"}],
Print[" "];
Print[" Wrong key word use: "];
Print[" Pcrit, Nc or Dim.
"];
Print[" "]
]
]
]
];
End[];
EndPackage[];
7.8.5 Fractional Calculus
Define the global variable $FractionalCalculusPath in such a way that the
location of the package FractionalCalculus is uniquely defined.
$FractionalCalculusPath =
$AddOnsDirectory <> "ЙApplicationsЙFracCalcЙ";
AppendTo@$Path, $FractionalCalculusPathD;
Load the package:
<< FractionalCalculus.m
--> FractionalCalculus ready <-╘ Gerd Baumann, Norbert SЭdland 1996-2004
<< Integral.m
-- "Integral.m" is available. --
NotebookClose@foxtitleD;
A
Appendix
This appendix contains some information on the installation of the
accompanying software and a short description of the functions defined in
the packages. It also summarizes the Mathematica functions used in the
book.
A.1 Program Installation
The book is accompanied by a CD containing all Mathematica notebooks.
These notebooks can be used as interactive text. The notebooks are linked
to a style file called ScriptStyle.nb and Vortrag.nb. You should copy these
two files to the location where the additional style files are stored. For
example,
on
a
PC,
the
style
files
are
located
at
C:\WINDOWS\Profiles\All_Users\Applications\Mathematica\SystemFiles\
FrontEnd\StyleSheets.
In addition to the notebooks, there is the package EulerLagrange which is
delivered with the text. The package is used in Chapter 2. In this chapter
you have to change the path name in the sections Packages and Programs.
900
1.1 Program Installation
You can either use the package from the CD or you can copy the package
to your preferred location. In any case, you have to change the path name
of the package.
Other packages supporting calculations of the text are located in the
section Packages and Programs in the appropriate notebook. For these
packages, there is no need to set any path names. They are ready to use for
your calculations.
A.2 Glossary of Files and Functions
This section contains a short description of all functions defined in the
packages of this book. The packages are alphabetically listed.
A.2.1 Anharmonic Oscillator
Anharmonic oscillator of quantum mechanics.
Х
AsymptoticPT
AsymptoticPT[N_,kin_] determines the asymptotic approximation for
╩ x |ь╤ for the continuous case of eigenvalues in a PЖschel?Teller
potential. The function yields an analytical expression for ╩ bHkL ╩2 . The
variables Transmission and Reflection contain the expressions for the
transmission and the reflection coefficients. w1a and w2a contain the
approximations for x ь -╤ and x ь ╤, respectively.
Х
PoeschelTeller
PoeschelTeller[x_, n_, N_] calculates the eigenfunction of the
PЖschel?Teller potential for discrete eigenvalues. N determines the depth
of the potential V0 SechHxL by V0 = NHN + 1L. n fixes the state where
0 < n ╖ N.
Х PlotPT
A. Appendix
901
PlotPT[kini_,kend_,type_] gives a graphical representation of the
reflection or transmission coefficient depending on the value of the
variable type. If type is set to the string r, the reflection coefficient is
plotted. If type is set to t, the transmission coefficient is represented. This
function creates five different curves.
Х Reflection
Variable containing the reflection coefficient. The independent variables
are N and k.
Х Transmission
Variable containing the expression for the transmission coefficient. The
independent variables are N and k.
Х w1a
The variable contains the analytic expression for the asymptotic
approximation for x ь -╤.
Х w2a
The variable contains the analytic expression for the asymptotic
approximation for x ь ╤.
A.2.2 Boundary Value Problem of Electrodynamics
Boundary value problem of electrodynamics.
Х Potential
Potential[boundary_,R_,alpha_,n_] calculates the potential in a circular
segment. Input parameters are the potential on the circle, the radius R of
the circle, and the angle of the segment of the circle. The last argument n
determines the number of expansion terms used to represent the solution.
902
1.2 Glossary
A.2.3 Central Field Problem in Quantum Mechanics
Quantum mechanical description of motion in a central field.
Х Angle
Angle[theta_, phi_, l_, m_] calculates the angular part of the wave function
for an electron in the Coulomb potential. The numbers L and m denote the
quantum numbers for the angular momentum operator. q and f are the
angles in the spherical coordinate system.
Х AnglePlot
AnglePlot[pl_,theta_,phi_] gives a graphical representation of the function
contained in pl. The range of representation is p ╖ f < 5 p Й 2 and 0 < q < p
. q is measured with respect to the vertical axis. This function is useful for
plotting the orbitals and the angular part of the eigenfunction.
Х Orbital
Orbital[theta_,phi_,l_,m_,type_String] calculates the superposition of two
wave functions for the quantum numbers ml = +m and ml = -m. The
variable type allows the creation of the sum or the difference of the wave
functions. The string values of type are either plus or minus.
Х Radial
Radial[ro_, n_, l_, Z_] calculates the radial representation of the
eigenfunctions for an electron in the Coulomb potential. The numbers n
and l are the quantum numbers for the energy and the angular momentum
operator. Z specifies the number of charges in the nucleus. The radial
distance between the center and the electron is given by r.
A.2.4 Harmonic Oscillator in Quantum Mechanics
Х a
A. Appendix
903
a[psi_, xi_:x] is annihilation operator for eigenfunction y. The second
argument specifies the independent variable of the function y.
Х across
across[psi_, xi_:x] is creation operator for eigenfunction y. The second
argument specifies the independent variable of y.
Х Psi
Psi[xi_,n_] represents the eigenfunction of the harmonic oscillator. The
first argument x is the spatial coordinate. The second argument n fixes the
eigenstate.
Х wcl
wcl[xi_,n_] calculates the classical probability of locating the particle in
the harmonic potential. The first argument x is the spatial coordinate and n
determines the energy given as the eigenvalue.
Х wqm
wqm[xi_,n_] calculates the quantum mechanical probability for an
eigenvalue state n. The first argument represents the spatial coordinate.
?
A.2.5 Korteweg?de Vries Equation
Multisoliton solution of the Korteweg?de Vries equation.
Х Soliton
Soliton[x_,t_,N_] creates the N soliton solution of the KdV equation.
Х PlotKdV
PlotKdV[tmin_,tmax_,dt_,N_] calculates a sequence of pictures for the N
soliton solution of the KdV equation. The time interval of the
904
1.2 Glossary
representation is @tmin , tmax D. The variable dt measures the length of the
time step.
A.2.6 Korteweg de Vries equation and its derivation
Derivation of the Korteweg de Vries equation.
Х Equation
Equation[n_] calculates the evolution equation up to order n.
A.2.7 Korteweg?de Vries Equation and Integrals of Motion
Integral of motion of the Korteweg?de Vries equation.
Х Gardner
Gardner[N_] calculates the densities of the integrals of motion for the KdV
equation using Gardner's method. The integrals are determined up to the
order N.
A.2.8 Korteweg?de Vries Equation Numerical Solution
Numerical solution of the Korteweg?de Vries equation.
Х KdVNIntegrate
KdVNIntegrate[initial_,dx_,dt_,M_] carries out a numerical integration of
the KdV equation using the procedure of [3.5]. The input parameter
initially determines the initial solution in the procedure (e.g., -6 Sech2 HxL).
The infinitesimals dx and dt are the steps with respect to the spatial and
temporal directions. M fixes the number of steps along the x-axis.
A. Appendix
905
A.2.9 Koch Curves
Fractal curves.
Х Fractal
Fractal[curve_String, options___] creates a graphical representation of a
fractal curve. The type of curve is determined by the first argument. A list
of available curves is obtained by calling Fractal[List] or Fractal[Help].
The second argument allows changing the options of the function. The
default values are Generations ь 3, Angle ь p Й 6 and Corners ь 6.
A.2.10 Light Beam Near a Planet
The bending of a light beam near a planet is discussed.
Х Deviation
Deviation[radius_,mass_] calculates the numerical value of the light
bending in a gravitational field of a star with mass M in a distance radius
of the center.
Х Orbit
Orbit[radius_,mass_] plots the orbit of a light beam near a mass in the
distance radius. The calculation is done in Schwarzschild metric.
A.2.11 Multifractal Properties
Multifractal properties of point sets.
Х MultiFractal
MultiFractal[p_List,r_List] calculates the multi-fractal spectrum Dq for a
model based on the probabilities p and the scaling factors r. This function
plots five functions tHqL, Dq HqL, aHqL, f HqL, and f HaL .
906
1.2 Glossary
A.2.12 Penning Trap
Motion of two ions in a Penning trap.
Х PenningCMPlot
PenningCMPlot[x0_,y0_,x0d_,y0d_,w_] gives a graphical represen- tation
of the center of mass motion for two ions in the Penning trap. The plot is
created for a fixed cyclotron frequency w in cartesian coordinates Hx, y, zL.
x0 , y0 , x0 d, and y0 d are the initial conditions for integration.
Х PenningI
PenningI[r0_,z0_,e0_,n_,l_,te_] determines the numerical solution of the
equation of motion for the relative components. To integrate the equations
of motion, the initial conditions r0 = rHt = 0L, z0 = zHt = 0L, and the total
energy e0 are needed as input parameters. The momentum with respect to
the r-direction is set to pr0 = 0. Parameters l and n determine the shape of
the potential. The last argument te specifies the endpoint of the integration.
A.2.13 Perihelion Shift
Perihelion shift of a planet.
Х AngularMomentum
AngularMomentum[minorAxes_,majorAxes_,mass_] calculates the angular momentum of a planet.
Х D0Orbit
D0Orbit[planet_String,phiend_,options___] plots the orbit in the case of
vanishing determinants (see text).
Х Energy
Energy[minorAxes_,majorAxes_,mass_] calculates the energy of a planet.
A. Appendix
907
Х orbit
orbit[phiend_,minorAxes_,majorAxes_,mass_]
creates
a
graphical
representation of the perihelion shift if the major and minor axes and the
mass are given.
Х Orbit
Orbit[planet_String] creates a graphical representation of the perihelion
shift for the planets contained in the database.
Х PerihelionShift
PerihelionShift[minorAxes_,majorAxes_,mass_] calculates the numeri- cal
value of the perihelion shift.
Х Planets
Planets[planet_String] creates a list of data for planets and planetoids
stored in the database of the package PerihelionShift. The database
contains the names of the planets, their major axes, their eccentricities, and
the mass of the central planet. Planets['List'] creates a list of the planets in
the data base. Planets['name'] delivers the data of the planet given in the
argument.
A.2.14 Point Charges
Fields of point charges.
Х EnergyDensity
EnergyDensity[coordinates_List] calculates the density of the energy for
an ensemble of point charges. The cartesian coordinates are lists in the
form of {{x,y,z,charge},{...},...}.
Х Field
908
1.2 Glossary
Field[coordinates_List] calculates the electric field for an ensemble of
point charges. The cartesian coordinates are lists in the form
{{x,y,z,charge},{...},...}.
Х FieldPlot
FieldPlot[coordinates_List,type_,options___] creates a contour plot for an
ensemble of point charges. The plot type (Potential, Field, or Density) is
specified as a string in the second input variable. The third argument
allows a change of the Options of ContourPlot and PlotGradientField.
Х Potential
Potential[coordinates_List] creates the potential of an assembly of point
charges. The cartesian coordinates of the locations of the charges are given
in the form of {{x,y,z,charge},{x,y,z,charge},...}.
A.2.15 Poisson Bracket
Canonical Poisson bracket.
Х PoissonBracket
PoissonBracket[a_, b_, q_List, p_List] calculates the Poisson bracket for
two functions a and b which depend on the variables p and q. Example:
PoissonBracket[q,p,{q},{p}] calculates the fundamental bracket relation
between the coordinate and momentum.
A.2.16 Quantum Well
Quantum well in one dimension.
Х PsiASym
PsiASym[x_,k_,a_] determines the antisymmetric eigenfunction for a
potential well of depth -V0 . The input parameter k fixes the energy and 2 a
A. Appendix
909
the width of the well. PsiASym is useful for a numerical representation of
eigenfunctions.
Х PsiSym
PsiSym[x_,k_,a_] determines the symmetric eigenfunction for a potential
well of depth -V0 . The input parameter k fixes the energy and 2 a the
width of the well. PsiSym is useful for a representation of eigenfunctions.
Х Spectrum
Spectrum[V0_,a_] calculates the negative eigenvalues in a potential well.
V0 is the potential depth and 2 a the width of the well. The eigenvalues are
returned as a list and are available in the variables lsym and lasym as
replacement rules. The corresponding eigenfunctions are stored in the
variables Plsym and Plasym. The determining equation for the
eigenvalues is plotted.
A.2.17 Renormalization
Renormalization and percolation.
Х Dim
Dim[n_] calculates the fractal dimension for the critical probability pc .
The dimension depends on m where 1 ╖ m ╖ n - 2, n is the size of the
block used.
Х Nc
Nc[n_] determines the mean number of atoms at the probability pc if m is
changed in the range 1 ╖ m ╖ n - 2. The size of the block is determined by
n.
Х Pcrit
910
1.2 Glossary
Pcrit[n_] determines the critical probability for an n Д n grid under the
variation of m where m is the number of empty locations in the grid. The
range of m is 1 ╖ m ╖ n - 2.
Х RenormPlot
RenormPlot[n_,type_String] plots the functions Nc, Dim or Pcrit.
A.2.18 Tree as a Fractal
Fractal tree.
Х Tree
Tree[options___] creates a fractal tree. The options of the function Tree
determine the form of the fractal created. Options are Generation ф 10,
BranchRotation ф 0.65, BranchSkaling ф 0.75, Branch- Thickness ф
0.7, OriginalThickness ф 0.07, BranchColor ф {RGBColor[0,0,0]}.
Example: Tree[BranchColor ф l1, BranchRotation ф 0.3], l1 is a list
created in the package Tree.
A.3 Mathematica Functions
This appendix contains a short description of the Mathematica functions
used in the book. It is a small selection of the approximately 1200
functions available in the Mathematica kernel. The description given does
not replace the text of the handbook by S. Wolfram ([1.1]).
The first few items describe the use of the shorthand notation of symbols
frequently used in the programming examples. The Mathematica functions
used in the programs and in the notebooks follow.
Х lhs = rhs evaluates rhs and assigns the result to lhs. From then on, lhs is
replaced by rhs whenever it appears. {l1, l2, ...}= {r1, r2, ...} evaluates
the ri and assigns the results to the corresponding li.
A. Appendix
911
Х lhs ф rhs represents a rule that transforms lhs to rhs.
Х expr /. rules applies a rule or list of rules to transform each subpart of an
expression expr.
Х lhs := rhs assigns rhs to be the delayed value of lhs. rhs is maintained in
an unevaluated form. When lhs appears, it is replaced by rhs, evaluated
afresh each time.
Х lhs :> rhs represents a rule that transforms lhs to rhs, evaluating rhs only
when the rule is used.
Х lhs == rhs returns True if lhs and rhs are identical.
Х expr //. rules repeatedly performs replacements
changes.
until expr no longer
Х AppendTo[s, elem] appends elem to the value of s and resets s to the
result.
Х Apply[f, expr] or f @@ expr replaces the head of expr by f. Apply[f,
expr, levelspec] replaces heads in parts of expr specified by levelspec.
Х ArcSin[z] gives the arc sine of the complex number z.
Х ArcTan[z] gives the inverse tangent of z. ArcTan[x, y] gives the inverse
tangent of y/x, where x and y are real, taking into account the quadrant in
which the point (x, y) is located.
Х Begin[ "context`"] resets the current context.
Х BeginPackage[ "context`"] makes context` and System` the only active
contexts. BeginPackage[ "context` ",{"need1` "}, { "need2` "},...}] calls
Needs on the needi.
Х BesselJ[n, z] gives the Bessel function of the first kind J(n, z).
Х Block[{x, y, ...}, expr] specifies that expr is to be evaluated with local
values for the symbols x, y, ... . Block[{x = x0, ...}, expr] defines initial
912
1.3 Mathematica Functions
local values for x,... . Block[{vars}, body /; cond] allows local variables to
be shared between conditions and function bodies.
Х C[i] is the default form for the ith constant of integration produced in
solving a differential equation with DSolve.
Х Chop[expr] replaces approximate real numbers in expr that are close to
zero by the exact integer 0. Chop[expr, tol] replaces with 0 approximate
real numbers in expr that differ from zero by less than tol.
Х Circle[{x, y}, r] is a two-dimensional graphics primitive that represents a
circle of radius r centered at the point {x, y}. Circle[{x, y}, {rx, ry}] yields
an ellipse with semiaxes rx and ry. Circle[{x, y}, r, {theta1, theta2}]
represents a circular arc.
Х Clear[symbol1, symbol2, ... ] clears values and definitions of the
specified symbols. Clear["pattern1", "pattern2", ...] clears values and
definitions of all symbols whose names match any of the specified string
patterns.
Х Coefficient[expr, form] gives the coefficient of form in the polynomial
expr. Coefficient[expr, form, n] gives the coefficient of formn in expr.
Х ContourPlot[f, {x, xmin, xmax}, {y, ymin, ymax}] generates a contour
plot of f as a function of x and y.
Х Cos[z] gives the cosine of z.
Х Cosh[z] gives the hyperbolic cosine of z.
Х Cot[z] gives the cotangent of z.
Х D[f, x] gives the partial derivative of f with respect to x. D[f, {x, n}] gives
the nth partial derivative with respect to x. D[f, x1, x2, ...] gives a mixed
derivative.
Х f' represents the derivative of a function f of one argument. Derivative[n1,
n2, ...][f] is the general form, representing a function obtained from f by
A. Appendix
913
differentiating n1 times with respect to the first argument, n2 times with
respect to the second argument, and so on.
Х Det[m] gives the determinant of the square matrix m.
Х Disk[{x, y}, r] is a two-dimensional graphics primitive that represents a
filled disk of radius r centered at the point {x, y}. Disk[{x, y}, 8rx , r y }]
yields an elliptical disk with semiaxes rx and rx . Disk[{x, y}, r, 8q1 , q2 }]
represents a segment of a disk.
Х Display[channel, graphics] writes graphics or sound to the specified
output channel.
Х Do[expr, {imax}] evaluates expr imax times. Do[expr, {i, imax}]
evaluates expr with the variable i successively taking on the values 1
through imax (in steps of 1). Do[expr, {i, imin, imax}] starts with i = imin.
Do[expr, {i, imin, imax, di}] uses steps di. Do[expr,{i, imin, imax}, {j,
jmin, jmax},... ] evaluates expr looping over different values of j, etc. for
each i. Do[] returns Null, or the argument of the first Return it evaluates.
Х DSolve[eqn, y[x], x] solves a differential equation for the functions y[x],
with independent variable x. DSolve[{eqn1, eqn2, ...},{y1[x1], ...}, {x1,
...}] solves a list of differential equations.
Х Dt[f, x] gives the total derivative of f with respect to x. Dt[f] gives the total
differential of f. Dt[f, {x, n}] gives the nth total derivative with respect to
x. Dt[f, x1, x2, ...] gives a mixed total derivative.
Х EllipticK[m] gives the complete elliptic integral of the first kind K(m).
Х End[ ] returns the present context, and reverts to the previous one.
Х EndPackage[ ] restores $Context and $ContextPath to their values before
the preceding BeginPackage, and prefixes the current context to the list
$ContextPath.
Х lhs == rhs returns True if lhs and rhs are identical.
914
1.3 Mathematica Functions
Х Evaluate[expr] causes expr to be evaluated, even if it appears as the
argument of a function whose attributes specify that it should be held
unevaluated.
Х Exp[z] is the exponential function.
Х Expand[expr] expands products and positive integer powers in expr.
Expand[expr, patt] avoids expanding elements of expr which do not
contain terms matching the pattern patt.
Х FindRoot[lhs == rhs, {x, x0 }] searches for a numerical solution to the
equation lhs == rhs, starting with x = x0 .
Х Flatten[list] flattens out nested lists. Flatten[list, n] flattens to level n.
Flatten[list, n, h] flattens subexpressions with head h.
Х Floor[x] gives the greatest integer less than or equal to x.
Х FontForm[expr, {"font", size}] specifies that expr should be printed in
the specified font and size.
Х Function[body] or body& is a pure function. The formal parameters are #
(or #1), #2, etc. Function[x, body] is a pure function with a single formal
parameter x. Function[{x1, x2,...}, body] is a pure function with a list of
formal parameters. Function[{x1, x2, ...}, body, {attributes}] has the given
attributes during evaluation.
Х <<name reads in a file, evaluating each expression in it, and returning the
last one. Get["name ", key] gets a file that has been encoded with a certain
key.
Х Graphics[primitives, options] represents a
image.
two-dimensional graphical
Х GraphicsArray[{g1, g2, ...}] represents a row of graphics objects.
GraphicsArray[{{g11, g12, ...}, ...}] represents a two-dimensional array of
graphics objects.
A. Appendix
915
Х HermiteH[n, x] gives the nth Hermite polynomial.
Х Hold[expr] maintains expr in an unevaluated form.
Х Hue[h] specifies that graphical objects which follow are to be displayed, if
possible, in a color corresponding to hue h. Hue[h, s, b] specifies colors in
terms of hue, saturation, and brightness.
Х If[condition, t, f] gives t if condition evaluates to True, and f if it
evaluates to False. If[condition, t, f, u] gives u if condition evaluates to
neither True nor False.
Х Im[z] gives the imaginary part of the complex number z.
Х Infinity is a symbol that represents a positive infinite quantity.
Х Input[ ] interactively reads in one Mathematica expression.
Input["prompt"] requests input, using the specified string as a prompt.
Х Integrate[f,x] gives the indefinite integral of f with respect to x.
Integrate[f,{x,
xmin,xmax}]
gives
the
definite
integral.
Integrate[f,{x,xmin,xmax},{y,ymin,ymax}] gives a multiple integral.
Х InterpolatingFunction[range, table] represents an approximate function
whose values are found by interpolation.
Х JacobiAmplitude[u, m] gives the amplitude for Jacobi elliptic functions.
Х JacobiSN[u, m] gives the Jacobi elliptic function sn at u for the parameter
m.
Х Join[list1, list2,... ] concatenates lists together. Join can be used on any set
of expressions that have the same head.
Х LaguerreL[n, x] gives the nth Laguerre polynomial. LaguerreL[n, a, x]
gives the nth generalized Laguerre polynomial.
Х LegendreP[n, x] gives the nth Legendre polynomial. LegendreP[n, m, x]
gives the associated Legendre polynomial.
916
1.3 Mathematica Functions
Х Length[expr] gives the number of elements in expr.
Х Limit[expr, x ф x0 ] finds the limiting value of expr when x approaches x0 .
Х Line[{pt1, pt2,...}] is a graphics primitive which represents a line joining a
sequence of points.
Х {e1, e2, ...} is a list of elements.
Х ListPlot[{y1, y2, ...}] plots a list of values. The x coordinates for each
point are taken to be 1, 2, ... . ListPlot[{{x1, y1}, {x2, y2}, ...}] plots a list
of values with specified x and y coordinates.
Х Log[z] gives the natural logarithm of z (logarithm to base E). Log[b, z]
gives the logarithm to base b.
Х Map[f, expr] or f /@ expr applies f to each element on the first level in
expr. Map[f, expr, levelspec] applies f to parts of expr specified by
levelspec.
Х MapAt[f, expr, n] applies f to the element at position n in expr. If n is
negative, the position is counted from the end. MapAt[f, expr, {i, j, ...}]
applies f to the part of expr at position {i, j, ...}. MapAt[f, expr, {{i1,
j1,...}, {i2, j2, ...}, ...}] applies f to parts of expr at several positions.
Х MatrixForm[list] prints the elements of list arranged in a regular array.
Х Max[x1, x2, ...] yields the numerically largest of the xi. Max[{x1, x2, ...},
{y1, ...}, ... ] yields the largest element of any of the lists.
Х Min[x1, x2, ...] yields the numerically smallest of the xi. Min[{x1, x2,
...}, {y1,...},...] yields the smallest element of any of the lists.
Х Mod[m, n] gives the remainder on division of m by n. The result has the
same sign as n.
Х N[expr] gives the numerical value of expr. N[expr, n] does computations
to n-digit precision.
A. Appendix
917
Х NDSolve[eqns, y, {x, xmin, xmax}] finds a numerical solution to the
differential equations eqns for the function y with the independent variable
x in the range xmin to xmax. NDSolve[eqns, {y1, y2,...}, {x, xmin,
xmax}] finds numerical solutions for the functions yi. NDSolve[eqns, y,
{x, x1, x2, ...}] forces a function evaluation at each of x1, x2, ... . The
range of numerical integration is from Min[x1, x2, ...] to Max[x1, x2,...].
Х Needs["context` ", "file"] loads file if the specified context is not already
in $Packages. Needs["context`"] loads the file specified by
ContextToFilename["context`"] if the specified context is not already in
$Packages.
Х Nest[f, expr, n] gives an expression with f applied n times to expr.
Х NestList[f, expr, n] lists the results of applying f to expr 0 through n times.
Х NIntegrate[f, {x, xmin, xmax}] gives a numerical approximation to the
integral of f with respect to x over the interval xmin to xmax.
Х Normal[expr] converts expr to a normal expression, from a variety of
special forms.
Х NSolve[eqns, vars] attempts to solve numerically an equation or set of
equations for the variables vars. Any variable in eqns but not vars is
regarded as a parameter. NSolve[eqns] treats all variables encountered as
vars above. NSolve[eqns, vars, prec] attempts to solve numerically the
equations for vars using prec digits precision.
Х Off[symbol::tag] switches off a message, so that it is no longer printed.
Off[s] switches off tracing messages associated with the symbols. Off[m1,
m2, ...] switches off several messages. Off[ ] switches off all tracing
messages.
Х On[symbol::tag] switches on a message, so that it can be printed. On[s]
switches on tracing for the symbol s. On[m1, m2, ...] switches on several
messages ma, m2, ... . On[ ] switches on tracing for all symbols.
Х ParametricPlot[{fx, fy}, {t, tmin, tmax}] produces a parametric plot with
x and y coordinates fx and fy generated as a function of t.
918
1.3 Mathematica Functions
ParametricPlot[{{fx, fy}, {gx, gy}, ...}, {t, tmin, tmax}] plots several
parametric curves.
Х ParametricPlot3D[{fx, fy, fz}, {t, tmin, tmax}]
produces a
three-dimensional space curve parameterized by a variable t which runs
from tmin to tmax. ParametricPlot3D[{fx, fy, fz}, {t, tmin, tmax}, {u,
umin, umax}] produces a three-dimensional surface parameterized by t and
u. ParametricPlot3D[{fx, fy, fz, s}, ...] shades the plot according to the
color specifications. ParametricPlot3D[{{fx, fy, fz}, {gx, gy, gz}, ...}, ...]
plots several objects together.
Х expr[[i]] or Part[expr, i] gives the ith part of expr. expr[[-i]] counts from
the end. expr[[0]] gives the head of expr. expr[[i, j, ...]] or Part[expr, i, j,
...] is equivalent to expr[[i]][[j]] ... . expr[[ {i1, i2, ...}]] gives a list of the
parts i1, i2, ... of expr.
Х Partition[list, n] partitions list into non-overlapping sublists of length n.
Partition[list, n, d] generates sublists with offset d. Partition[list, {n1, n2,
...}, {d1, d2, ...}] partitions successive levels in list into length ni sublists
with offsets di.
Х Pi is pi, with numerical value 3.14159... .
Х Plot[f, {x, xmin, xmax}] generates a plot of f as a function of x from xmin
to xmax. Plot[{f1, f2, ...}, {x, xmin, xmax}] plots several functions fi.
Х x + y + z represents a sum of terms.
Х Point[coords] is a graphics primitive that represents a point.
Х x y gives x to the power y.
Х PowerExpand[expr] expands nested powers, powers of products,
logarithms of powers, and logarithms of products. PowerExpand[expr,{x1,
x2,...}] expands expr with respect to the x1. Use PowerExpand with
caution because PowerExpand does not pay attention to branch cuts.
Х Print[expr1, expr2,... ] prints the expri, followed by a newline (line feed).
A. Appendix
919
Х Protect[s1, s2, ... ] sets the attribute Protected for the symbols si. Protect[
"form1", "form2 ", ...] protects all symbols whose names match any of the
string patterns formi.
Х Quit[ ] terminates a Mathematica session.
Х Random[ ] gives a uniformly distributed pseudorandom Real in the range
0 to 1. Random[type, range] gives a pseudorandom number of the
specified type, lying in the specified range. Possible types are Integer,
Real, and Complex. The default range is 0 to 1. You can give the range
{min, max} explicitly; a range specification of max is equivalent to {0,
max}.
Х Re[z] gives the real part of the complex number z.
Х ReleaseHold[expr] removes Hold and HoldForm in expr.
Х Replace[expr, rules] applies a rule or list of rules in an attempt to
transform the entire expression expr.
Х expr /. rules applies a rule or list of rules in an attempt to transform each
subpart of an expression expr.
Х expr //. rules repeatedly performs replacements until expr no longer
changes.
Х RGBColor[red, green, blue] specifies that graphical objects which follow
are to be displayed, if possible, in the color given.
Х lhs фrhs represents a rule that transforms lhs to rhs.
Х Save["filename", symb1, symb2, ...] appends the definitions of the
symbols symbi to a file.
Х Series[f, {x, x0 , n}] generates a power series expansion for f about the
point x = x0 to order Hx - x0 Ln . Series[f, {x, x0 , nx}, {y, y0 , ny}]
successively finds series expansions with respect to y, then x.
920
1.3 Mathematica Functions
Х Show[graphics, options] displays two- and three-dimensional graphics,
using the options specified. Show[g1, g2, ...] shows several plots
combined. Show can also be used to play Sound objects.
Х Simplify[expr] performs a sequence of transformations on expr and
returns the simplest form it finds.
Х Sin[z] gives the sine of z.
Х Sinh[z] gives the hyperbolic sine of z.
Х Solve[eqns, vars] attempts to solve an equation or set of equations for the
variables vars. Any variable in eqns but not vars is regarded as a
parameter. Solve[eqns] treats all variables encountered as vars above.
Solve[eqns, vars, elims] attempts to solve the equations for vars,
eliminating the variables elims.
Х Sort[list] sorts the elements of list into canonical order. Sort[list, p] sorts
using the ordering function p.
Х SphericalHarmonicY[l, m, theta, phi] gives the spherical harmonic
Yl,m (q, f).
Х Sqrt[z] gives the square root of z.
Х Sum[f, {i, imax}] evaluates the sum of f with i running from 1 to imax.
Sum[f, {i, imin, imax}] starts with i = imin. Sum[f, {i, imin, imax, di}]
uses steps di. Sum[f, {i, imin, imax}, {j, jmin, jmax},...] evaluates a
multiple sum.
Х Table[expr, {imax}] generates a list of imax copies of expr. Table[expr,
{i, imax}] generates a list of the values of expr when i runs from 1 to imax.
Table[expr, {i, imin, imax}] starts with i = imin. Table[expr, {i, imin,
imax, di}] uses steps di. Table[expr, {i, imin,
imax}, {j, jmin,
jmax},...] gives a nested list. The list associated with i is outermost.
Х Take[list, n] gives the first n elements of list. Take[list, -n] gives the last n
elements of list. Take[list, {m, n}] gives elements m through n of list.
A. Appendix
921
Х Tan[z] gives the tangent of z.
Х Text[expr, coords] is a graphics primitive that represents text
corresponding to the printed form of expr, centered at the point specified
by coords.
Х Thread[f[args]] ``threads'' f over any lists that appear in args.
Thread[f[args], h] threads f over any objects with head h that appear in
args. Thread[f[args], h, n] threads f over objects with head h that appear in
the first n args. Thread[f[args], h, -n] threads over the last n args.
Thread[f[args], h, {m, n}] threads over arguments m through n.
Х Unprotect[s1, s2, ...] removes the attribute Protected for the symbols si.
Unprotect["form1","form2", ...] unprotects all symbols whose names
textually match any of the formi.
Х Which[test1, value1, test2, value2, ... ] evaluates each of the testi in turn,
returning the value of the valuei corresponding to the first one that yields
True.
References
Volume I
[1]
Chapter 1
[1.1]
S. Wolfram, The Mathematica book, 5th ed.
Media/Cambridge University Press, Cambridge 2003.
[1.2]
M. Abramowitz & I.A. Stegun, Handbook of Mathematical
Functions. Dover Publications, Inc., New York, 1968.
[1.3]
N. Blachman, Mathematica: A Practical Approach. Prentice Hall,
Englewood Cliffs, 1992.
[1.4]
Ph. Boyland, A. Chandra, J. Keiper, E. Martin, J. Novak, M.
Petkovsek, S. Skiena, I. Vardi, A. Wenzlow, T. Wickham-Jones,
D. Withoff, and others, Technical Report: Guide to Standard
Mathematica Packages, Wolfram Research, Inc. 1993.
[2]
Chapter 2
Wolfram
924
References
[2.1]
R. Maeder, Programming in Mathematica. Addison-Wesley Publ.
Comp. Inc., Redwood City, 1991.
[2.2]
L.D. Landau & E.M. Lifshitz, Mechanics. Addison-Wesley,
Reading, Massachusetts, 1960.
[2.3]
J. B. Marion, Classical Dynamics of Particles and Systems.
Academic Press, New York, 1970.
[2.4]
R. Courant & D. Hilbert, Methods of Mathematical Physics, Vol.
1+2. Wiley (Interscience), New York, 1953.
[2.5]
R.H. Dicke, Science 124, 621, (1959)
[2.6]
R.V. EЖtvЖs,Ann.Phys. 59, 354, (1896)
[2.7]
L. Southerns,Proc.Roy.Soc.(London),A, 84, 325, (1910)
[2.8]
P. Zeeman,Proc.Amst.,20,542,(1917)
[2.9]
G. Baumann, Symmetry Analysis of Differential equations using
Mathematica, Springer, New York, (2000).
[2.10]
H. Geiger and E. Marsden, The Laws of Deflexion of a Particles
through Large Angles, Phil. Mag. 25, 605, 1913.
[2.11]
Ph. Blanchard and E. BrЭning, Variational Methods in
Mathematical Physics, Springer, Wien, 1982.
[3]
Chapter 3
[3.1]
F. Calogero & A. Degasperis, Spectral Transform and Solitons:
Tools to solve and investigate nonlinear evolution equations.
North-Holland Publ. Comp., Amsterdam, 1982.
[3.2]
V.A. Marchenko, On the Reconstruction of the Potential Energy
from Phases of the Scattered Waves. Doklady Akademii Nauk
SSSR, 104, 695, 1955.
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R.M. Miura, C. Gardner & M.D. Kruskal. Korteweg-de Vries
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[3.4]
T.R. Taha & M.J. Ablowitz, Analytical and numerical solutions of
certain nonlinear evolution equations. I. Analytical. Journal of
Computational Physics 55, 192, 1984.
[3.5]
N.J. Zabusky & M.D. Kruskal, Interactions of 'solitons' in a
collisionless plasma and the recurrence of initial states. Physical
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Volume II
[4]
Chapter 4
[4.1]
G. Arfken, Mathematical Methods for Physicists. Academic Press,
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[4.2]
P.M. Morse & H. Feshbach, Methods of Theoretical Physics.
McGraw-Hill, New York, 1953.
[4.3]
W. Paul, O. Osberghaus & E. Fischer, Ein IonenkДfig.
Forschungsbericht des Wissenschafts- und Verkehrsministeriums
Nordrhein-Westfalen, 415, 1, 1958.
Similar work has been done by H. G. Dehmelt, Radiofrequency
Spectroscopy of Stored Ions I: Storage, Advances in Atomic and
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R.S. van Dyck Jr., High-Resolution Spectroscopy of Stored Ions,
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F.M. Penning, Die Glimmentladung bei niedrigem Druck zwischen
koaxialen Zylindern in einem axialen Magnetfeld. Physica 3, 873,
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[4.5]
G. Baumann, The Paul trap: a completely integrable model? Phys.
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Chapter 5
[5.1]
E. SchrЖdinger, Quantisierung als Eigenwertproblem. Annalen der
Physik, 79, 361, 1926.
[5.2]
N. Rosen & P.M. Morse, On the Vibrations of Polyatomic
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[5.3]
G. PЖschel & E. Teller, Bemerkungen zur Quantenmechanik des
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[5.4]
W. Lotmar, Zur Darstellung des Potentialverlaufs
zweiatomigen MolekЭlen. Z. Physik, 93, 518, 1935
[5.5]
S. FlЭgge, Practical Quantum Mechanics I + II. Springer-Verlag,
Berlin, 1971.
[5.6]
C. Cohen-Tannoudji, B. Diu & F. LaloК, Quantum Mechanics I +
II. John Wiley & Sons, New York, 1977.
[5.7]
Rowlinson J.S.; Mol. Phys. 1963, 6, 75-83
[5.8]
Lennard-Jones J.E.; Proc. Roy. Soc. 1924, A106, 463-477
[5.9]
London F.; Z. Phys. 1930, 63, 245-279
bei
[5.10]
Hirschfelder J.O., Curtiss R.F., Bird R.B. Molecular Theory of
Gases and Liquids. Wiley: New York, 1954
[5.11]
Mason E.A., Spurling T.H. The virial Equation of State;
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[5.12]
McQuarrie D.A.; Statistical Thermodynamics, Harper and Row:
New York 1973, p. 307
[5.13]
Sinanoglu O. and Pitzer K.S.; J. Chem. Phys. 1959, 31, 960-967
[5.14]
Friend D.G.; J. Chem. Phys. 1985, 82, 967-971
[5.15]
Kihara T.; Suppl. Progs. Theor. Phys. 1967, 40, 177-206
[5.16]
Stogryn D.E., Hirschfelder J.O. J. Chem. Phys. 1959, 31,
1531-1545
[5.17]
Phair R., Biolsi L., Holland P.M. Int. J. Thermophys., 1990, 11,
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Mies F.H., Julienne P.S. J. Chem. Phys. 1982, 77, 6162-61176
[6]
Chapter 6
[6.1]
W. Rindler, Essential Relativity. Springer Verlag, New York, 1977.
[6.2]
C.W. Misner, K.S. Thorne & J.A. Wheeler, Gravitation. Freeman,
San Francisco, 1973.
[6.3]
H. Stephani, General relativity: An introduction to the gravitational
field. Cambridge University Press, 1982.
[6.4]
M. Berry, Principles of Cosmology and Gravitation. Cambridge
University Press, Cambridge, 1976.
[7]
Chapter 7
[7.1]
T.W. Gray & J. Glynn, Exploring Mathematics with Mathematica.
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[7.2]
T.F. Nonnenmacher, G. Baumann & G. Losa, Self organization and
fractal scaling patterns in biological systems. In: Trends in
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A. Barth, G. Baumann & T.F. Nonnenmacher, Measuring
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[7.4]
B. Mandelbrot, The fractal geometry of nature. W.H. Freeman a.
Comp., New York, 1983.
[7.5]
A. Aharony, Percolation. In: Directions in condensed matter
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[7.6]
T. Grossman & A. Aharony, Structure and perimeters of
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Index
A
Abel, 941
absolute temprature, 766
ac-field, 610
action, 779
algebraic equation, 986
algorithm, 987, 993
amorphous semiconductor, 997
amplitude, 731
analytical calculation, 545
analytical methods, 906
angle of inclination, 793
angular momentum, 616, 751?752,
786
angular quantum number, 757
anharmonic, 740
anharmonic oscillator, 740
anhilation operator, 738
annihilation operator, 737
anomalous diffusion, 984, 1006
anomalous diffusion exponent, 1006
ansatz, 755
aphelion, 783
apogee, 789
associated Legendre polynomials, 741
assumption, 949
astrophysics, 807
asymptotic circles, 789
asymptotic direction, 794
asymptotic expansion, 747
asymptotic representation, 748
atomic systems, 706
average energies, 803
Avogadro number, 767
Avogadro's constant, 766
axial frequency, 613
B
balls, 903
Barns integral, 983
base angle, 920
Bernoulli, 939
Bessel function, 956
932
Bianchi identities, 803, 811
binding of atoms, 758
black hole, 706
blackbody radiation, 703
blocks, 931
Boltzmann constant, 766?767
borderline, 903
Born, 705
bound region, 803
bound state, 768, 803
boundary, 900
boundary condition, 590
Dirichlet, 600
Dirichlet and von Neumann,
600
von Neumann, 600
boundary line, 905
boundary problem, 598?599
bounded sets, 900
bounded subset, 908
box counting, 906, 908
box counting dimension, 908
box counting method, 905
box dimension, 908, 912
box length, 914
Boyle temperature, 803
Boyle temperaure, 805
Broglie, 704
bronchial tree, 905
C
calculus, 948
Index
Cantor, 906
capacity dimension, 908
Cartesian coordinates, 592
Cartesian metric, 797
Cartesian space, 804
Cauchy's integral formula, 942
center of mass coordinates, 611
center of mass motion, 612
central field, 752
central force, 777
central force field, 751
chain rule, 945, 947
changing scales, 930
chaotic, 617
characteristic function, 924
characteristic polynomial, 613, 783,
792
charge density, 590
charge distribution, 590
charge-free, 600
charged mass point, 822
Christoffel symbols, 801, 805
circular force, 588
classical mechanics, 546, 715
classical orbit, 789
classical probability, 733
classically forbidden, 715
commuting operators, 752
complete basis, 713
complete elliptic integrals, 787
complex field, 707
Index
complex materials, 997
composition rule, 945?946
conducting wall, 609
cones, 903
confluent hypergeometric function,
756
congruence, 919
congruent triangle, 918
continuity condition, 716
continuum state, 768, 803
continuum theory, 599
contour length, 908
contour plot, 592
convolution, 961, 963
convolution type integral, 974
coordinate transformation, 804
correlation length, 935
Coulomb, 588
Coulomb force, 611
Coulomb interaction, 611, 754
count, 912
countable sets, 900
covariant divergence, 823
creation operator, 737
critical exponent, 935?936
critical phenomena, 930
critical point, 930, 935
curvature scalar, 802
curved space, 774?775
cyclotron frequency, 613, 616
cylinders, 903, 908
933
cylindrical coordinates, 806
cylindrical coordinates , 798
D
Davy, 588
dc-potential, 612
Debye process, 995
Debye relaxation, 995
decades, 997
degenerate electronic states, 808
density, 734
derivatives, 963
determinant, 717
diagonal elements, 810
diatomic molecule, 740, 808
diatomic molecules, 807
dielectric relaxation, 997
differential equation, 985?986
differential equations, 964
differentiation of a constant, 949
diffusion constant, 707, 1007
diffusion equation, 707
dimer parition function, 808
Dingle's metric, 812
dipole, 592
Dirac's delta function, 590
Dirichlet boundary condition, 600
Dirichlet problem, 600
discrete spectrum, 602, 745
disjunct boxes, 908
disociation limit, 809
934
dispersion, 708, 712
dispersion force, 767
dispersion relation, 712
dispersive phenomena, 709
dispersive wave, 708
distribution, 972
domain boundaries, 716
driven rubber equation, 1004
dynamic trap, 609
dynamo, 588
E
eccentricity, 786
Eddington-Finkelstein, 809
Eddington-Finkelstein line element,
809
edge length, 909
eigenfunction, 601, 713, 731?732,
739, 743
antisymmetric, 718
symmetric, 718
eigenfunction expansion, 601
eigenstate, 713
eigenvalue, 601, 713, 715
eigenvalue equation, 720
eigenvalue problem, 601, 731, 752
eikonal equation, 707
Einstein tensor, 819
Einstein's field equation, 773
Einstein's field equations, 795, 799,
803
electric field, 590?591
electric force, 588
electric potential, 600
Index
electricity, 588
electromagnetic field, 589
electromagnetic force, 611
electromagnetic phenomena, 590
electronic degeneracy, 808
electrostatic, 590
electrostatic phenomena, 599
ellipse, 777
ellipsoids, 908
elliptic function, 780
energy, 714, 786
energy density, 777
enthalpy, 768, 778
entropy, 768, 778
entropy dimension, 908
equation of state, 769
equilibrium point, 730
Euclidean space, 797
Euler, 941
Euler-Lagrange equations, 779
excitation energy, 808
expansion coefficient, 601
expectation value, 934
exponential, 987
exponential decay, 996
external force, 989
external potential, 707
F
Farady, 588
field, 588
Index
field equations, 801
first formula by Green, 599
first kind Fredholm integral equation,
976
first quantum correction, 780
fit, 916
fixed point, 932
flat space, 805
FlЭgge, 740
focus, 777
Fourier, 941
Fourier transform, 708, 958, 1008
Fox H-function, 968
Fox function, 967, 982?983
fractal, 906, 930
fractal cluster dimension, 935
fractal dimension, 906
fractal geometry, 937
fractals, 546
Fractals, 899
fractional calculus, 937
fractional derivative, 943
fractional derivatives, 940, 943
fractional differential equations, 984
fractional differentiation, 937, 943,
949
fractional dimension, 900
fractional integral, 953
fractional integral equation, 959
fractional relaxation equation, 995
FractionalCalculus, 949
Fredholm convolution integral, 972
935
Fredholm equation, 973
Fredholm integral equation, 979, 998
free particle, 709
Friedman, 774
fundamental force, 706
G
G-function, 939, 964
gas, 930
gas constant, 766
gas imperfection, 769
gauge conditions, 804
Gauъ, 938
Gaussian behavior, 1006
Gaussian coordinates, 804
Gauss's law, 590
Gauss's theorem, 599
general relativity, 773
generalized diffusion equation, 1007
generalized dimension, 924, 926
generalized hypergeometric function,
967
generalized Mittag-Leffler function,
998
generalized relaxation equation, 991
generating operator, 737
geometric complexity, 900
geometric mass, 827
geometric structure, 899
geometrical objects, 903
Gibb's techniques, 766
gravitation, 599
gravitation phenomena, 775
gravitational collapses, 774
936
gravitational constant, 778
gravitational field, 777
gravitational radiation, 774
Green's, first formula, 600
second formula, 600
Green's function, 590, 599, 605, 708
ground electronic state, 809
ground state, 737
H
H-atom, 751
Hamiltonian, 730, 751
Hamiltonian operator, 714
Hankel transform, 959
harmonic external force, 1004
harmonic function, 613
harmonic oscillations, 730
harmonic oscillator, 613, 712, 729
Hausdorff, 900
heat capacity, 778
Heisenberg, 705
Hermite, 732
Hermite polynomial, 732, 737
high frequency limit, 703
high temperature chemistry, 807
HЖlder exponent, 925?926
hydrodynamics, 599
hydrogen atom, 755
hyper-geometric function, 745
hypergeometric function, 732, 772,
952
hypergeometric functions, 793
Index
I
induction, 588
information dimension, 908
inhomogeneous field equations, 822
initial condition, 708, 1007
initial value problem, 986?987
integral equation, 973, 975, 990
integral equations, 964, 972
integral theorem of Gauss, 600
integral transform, 958, 991
integral transforms, 986
intermolecular force, 771
intermolecular potential, 766
internal erenrgy, 774
internuclear distance, 769
invariant, 930
inverse metric tensor, 808
inverse scattering method, 740
inverse temperature, 772
InverseMellinTransform[], 966
ion trap, 609
isotropic, 800
J
Jones, 767
Jordan, 705
Joul-Thomson coefficient, 778
K
Kannerligh Onnes, 765
Kepler, 777, 789
kernel, 959, 975
Index
Kerr solution, 827
Kihara potential, 769?770
Koch, 906
Koch curve, 918?919
Koch snowflake, 906
Kohlrausch-William-Watts, 971
Kolmogorov entropy, 908
Kruskal coordinates, 818
Kruskal solution, 818
Kruskal variables, 822
Kummer's differential equation, 756
Kummer's function, 757
L
Lacroix, 941
Lagrangian, 617, 778
Laguerre polynomial, 757
Laguerre's function, 757
Langevin equation, 985
Laplace equation, 598, 609
cylindrical coordinates, 603
Laplace integral equation, 978
Laplace space, 987
Laplace transform, 771, 959,
986?987, 991
large molecule, 740
lattice, 931
Lebesgue, 900
Lebesgue measure, 900
Legendre function, 743, 753
Legendre polynomial, 741
Legendre transform, 925
937
Leibniz, 938
Leibniz rule, 945
Leibniz's rule, 947
length, 920
length of a border, 899
Lennard, 767
Lennard-Jones potential, 767, 769
Lenz vector, 777
L`Hospital, 938
light bending, 790
light ray, 790
light rays, 791
line element, 795, 804, 920
linear displacement, 740
linear first-order ODE, 985
linear fractional differential equation,
990
linearity, 708, 945, 990
Liouville, 939, 942
Liouville fractional integral, 943
liquid, 930
local minimum, 729
log-log plot, 906, 909
London, 767
Lorentz force, 611
Lotmar, 740
low frequency limit, 703
M
macroscopic thermodynamics, 765
magnetic field, 610
magnetic force, 588
938
magnetic quantum number, 753
magnetism, 588
major semi axis, 786
Mandelbrot, 899, 925
Mandelbrot set, 901
mapping, 901
mass density, 777
mathematical calculation, 545
matrix algebra, 705
matrix mechanics, 705
Maxwell, 588
Maxwell tensor, 823
Maxwell's equations, 822
mean square displacement, 1006
mean value, 707
measurement, 713
Meijer G-function, 968
Mellin representation, 994
Mellin space, 975, 992
Mellin transform, 958?960, 973, 975,
979, 991
Mellin-Barns integral, 994
MellinTransform[], 961
memory, 998
memory kernel, 1007
memory-diffusion equation, 1007
Mercury, 777, 785
mesh-size, 905, 934
metastable state, 768, 803
metric, 795
metric dimension, 908
Index
metric geodesics, 801
metric tensor, 795, 798?799, 801
microscopic physics, 765
Minkowski space, 799
Mittag-Leffler function, 952, 993
modulus, 794
molecular interactions, 766
molecular orbital, 758
molecular potential, 803
moments, 972
momentum space, 737
monoatomic assembly, 769
monomer partition function, 808
monster curves, 899
movement of perihelion, 775
multi-fractal, 924, 926
multi-fractal characteristic, 926
multi-fractal distribution, 925
multi-Fractals, 923
N
nth-order ODE, 985
nano phenomena, 706
natural objects, 899, 905
negative second-order derivative, 942
Newton, 611, 775, 777, 938
non-commutative algebra, 705
non-degenerate, 733
non-integer derivatives, 938
nonlinear evolution equation, 740
normal gradient, 600
Index
normalization, 716
normalize, 709
normalized solution, 752
null geodesic, 790
O
option, 951
orbit, 780
orbital, 764
orbital motion, 777
Ornstein, 766
orthogonal, 601
P
paraboloid, 609
parameterized curve, 801
partition function, 768, 807
Paul, 609
Peano, 906
Penning, 609
Penning trap, 609
percolation cluster, 931?932
percolation theory, 931
perfect gas, 768
perihelion, 777, 783
perihelion rotation, 777
perihelion shift, 777, 785
period, 730, 783
perturbation theory, 936
phase diagram, 930
phase transition, 932
phase transitions, 930
939
physical characteristics, 900
Planck, 703
Planck constant, 707
plane filling, 906, 921
plane wave, 708
planetary system, 777
point charge, 591
Poisson equation, 590
polymer, 984
polymer science, 931
polynomial, 732
porous medium, 931
PЖschel, 740
PЖschel-Teller potential, 740
potential, 590?591
potential barrier , 734
potential depth, 743
potential well, 714
power law, 937, 997
pressure, 803
pressure equilibrium constant, 808
principal quantum number, 757
probability, 707, 923
probability amplitude, 705
probability distribution, 710, 733
projection plane, 904
properties of the Mellin transform,
960
Pythagoras, 918
Q
quadruple, 595
940
quadrupole field, 609, 611
quantum chemistry, 740
quantum correction, 767, 778
quantum corrections, 767
quantum dot, 751
quantum dot model, 707
quantum mechanical corrections, 778
quantum mechanical operators, 731
quantum mechanical state, 737
quantum mechanics, 546, 704, 707
quantum number, 753, 757, 807
quasi elliptic orbits, 783
Index
relaxation equation, 986, 989
relaxation of polymers, 997
relaxation oscillation equation, 1000
relaxation phenomenon, 984
relaxation time, 986
relaxation time spectrum, 899
renormalization, 930
renormalization error, 936
renormalization group, 929?930
renormalized lattice, 931
repulsive branch, 804
resolution transformation, 929
rest mass, 777
Ricci scalar, 802?803
R
radial quantum number, 757
Ricci scalar , 825
radial wave function, 754
Ricci tensor, 801?803
random force, 985
Riemann, 775, 939, 942
random links, 931
Riemann fractional integral, 943
random number, 909
Riemann geometry, 795
rational function, 964
Riemann tensor, 801?802
Rayleigh, 703
Riemann tensor , 807
reaction kinetics, 807
Riemann z-function, 965
real gas, 766
Riemann-Liouville fractional integral,
reduced de Broglie wavelength, 789
943
reduced mass, 807
Riemann-Liouville operator, 945
reduced quantities, 793
RiemannLiouville[], 948
reflection coefficient, 747
RiemannLiouville[], 944
regularity, 604
Riemann's theory, 774
Reissner-Nordstrom solution, 773, 822 rosette, 784
relative coordinates, 611
rosettes, 777
relative motion of the ions, 615
rotating black hole, 827
Index
rotation-vibration eigenfunction, 807
rotation-vibration SchrЖdinger
equation, 807
rotational barrier, 807
Rydberg-diatomic potential, 768
S
scaling, 616, 731, 961
scaling behavior, 918
scaling exponent, 909, 916
scaling factor, 920, 926
scaling factors, 923
scaling property, 962
scaling range, 909
scaling transformation, 930
scattering problem, 748
SchrЖdinger, 704
SchrЖdinger equation, 707, 740, 752
Schwarzschild, 774
Schwarzschild line element, 810
Schwarzschild metric, 778, 790
Schwarzschild radius, 778, 791
Schwarzschild solution, 773, 799, 809
second formula by Green, 600
second kind of Fredholm equation,
979
second quantum correction, 780
second virial coefficient, 765?766,
769, 793
secular equation, 617
self-similar, 909
self-similarity, 903, 906, 918, 923
semi fractional derivative, 957
semi-group, 930
941
semiclassical expansion, 767
semiconductors, 706
semifractional differential equation,
1002
separation, 604
shifting, 961
shifting property, 962
singular, 810
singularity, 783
slope, 906
slow decay, 1000
small oscillations, 730
snowflake, 900
space time, 795
specific heat, 768
spectral density, 708, 712
spectral properties, 712
spectroscopic dissociation energy,
809
spectrum, 926
spheres, 908
spherical coordinates, 798, 807
spherical Einstein equations, 775
spherical symmetry, 799, 809, 822
spherically symmetric, 751
spring constant, 730
standard diffusion, 1007
standard relaxation, 995
static magnetic field, 611
static trap, 609
stationary SchrЖdinger equation, 745
statistical physics, 599
942
straight line, 903
straight lines, 903
super lattice, 931, 934
superposition, 707?708, 764, 945, 991
symmetric
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